One Mixed Nash Equiilbria for von Neumann Poker with number of cards from 2 to, 40, and bet sizes from, 1, to, 10 By Shalosh B. Ekhad Player 1 and Player 2 each get (different) cards from a deck of n cards numb\ ered 1,..., n. They each see their own card, but not the other's They each put 1 dollar in the pot Player 1 can either check, in which case the cards are compared, and whoever\ has the larger card gets the pot (and hence wins 1 dollar), or bet an a\ dditional amount of b dollars Now Player 2 can decide to cut his losses, and fold, and the pot goes immedi\ ately to player 1, who is going to make a dollar, and player 2 is going \ to lose a dollar or else to also put b dollars in the pot Now the cards are compared, and whoever has the larger card gets the pot, wi\ nning b+1 dollars (and the other player loses b+1 dollars Here is ONE Mixed one, for card sizes from 2 to, 40, and bet sizes from, 1, to, 10 For each number of cards n, and bet size, we list: the Player 1 strategy, as a list of length n, let's call it, P1, telling th\ e player that if he got card number x, he should bet with prob. P1[x] an\ d check otherwise. the Player 2 strategy, as a list of length n, let's call it, P2, telling th\ e player that if he got card number y, he should call with prob. P2[y] a\ nd fold otherwise. We also list the value, that we call V -------------------------------------------- The deck has, 2, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0], [0, 1]] -------------------------------------------- The deck has, 3, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/18, .5555555556e-1, [1/3, 0, 1], [0, 1/3, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0], [0, 0, 1]] -------------------------------------------- The deck has, 4, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1/3, 0, 0, 1], [0, 0, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1/2, 0, 0, 1], [0, 0, 1/2, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [1/20, .5000000000e-1, [3/5, 0, 0, 1], [0, 0, 1/5, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0], [0, 0, 0, 1]] -------------------------------------------- The deck has, 5, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1/3, 0, 0, 0, 1], [0, 0, 2/3, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1/2, 0, 0, 0, 1], [0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [9/100, .9000000000e-1, [3/5, 0, 0, 0, 1], [0, 0, 0, 3/5, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [1/15, .6666666667e-1, [2/3, 0, 0, 0, 1], [0, 0, 0, 1/3, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [1/28, .3571428571e-1, [5/7, 0, 0, 0, 1], [0, 0, 0, 1/7, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0], [0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0], [0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0], [0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0], [0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0], [0, 0, 0, 0, 1]] -------------------------------------------- The deck has, 6, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [4/45, .8888888889e-1, [2/3, 0, 0, 0, 1, 1], [0, 0, 1/3, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1/2, 0, 0, 0, 0, 1], [0, 0, 0, 1/2, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [3/5, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [4/45, .8888888889e-1, [2/3, 0, 0, 0, 0, 1], [0, 0, 0, 0, 2/3, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [1/14, .7142857143e-1, [5/7, 0, 0, 0, 0, 1], [0, 0, 0, 0, 3/7, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/20, .5000000000e-1, [3/4, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1/4, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [7/270, .2592592593e-1, [7/9, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1/9, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1]] -------------------------------------------- The deck has, 7, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [2/21, .9523809524e-1, [2/3, 0, 0, 0, 0, 1, 1], [0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [2/21, .9523809524e-1, [1/2, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [3/5, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 2/5, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [2/21, .9523809524e-1, [2/3, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [25/294, .8503401361e-1, [5/7, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 5/7, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/14, .7142857143e-1, [3/4, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1/2, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [1/18, .5555555556e-1, [7/9, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1/3, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [4/105, .3809523810e-1, [4/5, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1/5, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [3/154, .1948051948e-1, [9/11, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1/11, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [0, 0., [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1]] -------------------------------------------- The deck has, 8, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [2/21, .9523809524e-1, [2/3, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 2/3, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [3/28, .1071428571, [1, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 1/2, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [27/280, .9642857143e-1, [3/5, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 4/5, 1, 1] ] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [2/21, .9523809524e-1, [2/3, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1/3, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [5/56, .8928571429e-1, [5/7, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [9/112, .8035714286e-1, [3/4, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 3/4, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [5/72, .6944444444e-1, [7/9, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 5/9, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [2/35, .5714285714e-1, [4/5, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 2/5, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [27/616, .4383116883e-1, [9/11, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 3/11, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [5/168, .2976190476e-1, [5/6, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 1/6, 1]] -------------------------------------------- The deck has, 9, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [7/72, .9722222222e-1, [1, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 1/3, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [7/72, .9722222222e-1, [1, 0, 0, 0, 0, 0, 0, 2/3, 1], [0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [5/54, .9259259259e-1, [2/3, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 2/3, 1 , 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [5/56, .8928571429e-1, [5/7, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 2/7, 1 , 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [3/4, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [49/648, .7561728395e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 7 /9, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [1/15, .6666666667e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 3/5 , 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [5/88, .5681818182e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 5/ 11, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [5/108, .4629629630e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1/ 3, 1]] -------------------------------------------- The deck has, 10, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 1/2, 1, 1, 1 , 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [23/225, .1022222222, [1, 1/5, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [4/45, .8888888889e-1, [2/3, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [11/126, .8730158730e-1, [5/7, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0 , 4/7, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [3/4, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [7/90, .7777777778e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [16/225, .7111111111e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0 , 0, 4/5, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [7/110, .6363636364e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0 , 0, 7/11, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [1/18, .5555555556e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1]] -------------------------------------------- The deck has, 11, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [6/55, .1090909091, [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [29/275, .1054545455, [1, 1/5, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0 , 3/5, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [1/11, .9090909091e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [13/154, .8441558442e-1, [5/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0 , 0, 0, 6/7, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [9/110, .8181818182e-1, [3/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [7/90, .7777777778e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 2/9, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [4/55, .7272727273e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [81/1210, .6694214876e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 9/11, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [2/33, .6060606061e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1]] -------------------------------------------- The deck has, 12, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [13/132, .9848484848e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [7/66, .1060606061, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [7/66, .1060606061, [1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [19/198, .9595959596e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0 , 0, 0, 0, 1/3, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [7/88, .7954545455e-1, [3/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [91/1188, .7659932660e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4/9, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [4/55, .7272727273e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [3/44, .6818181818e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [25/396, .6313131313e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 5/6, 1]] -------------------------------------------- The deck has, 13, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [23/234, .9829059829e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0 , 0, 1/3, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [17/156, .1089743590, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0 , 0, 0, 1/2, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [41/390, .1051282051, [1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0 , 0, 0, 0, 2/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [23/234, .9829059829e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0 , 0, 0, 0, 0, 0, 2/3, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [47/546, .8608058608e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0 , 0, 0, 0, 0, 0, 1/7, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/13, .7692307692e-1, [3/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [35/468, .7478632479e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 2/3, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [14/195, .7179487179e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 2/5, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [3/44, .6818181818e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/11, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [5/78, .6410256410e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1]] -------------------------------------------- The deck has, 14, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [9/91, .9890109890e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [10/91, .1098901099, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [47/455, .1032967033, [1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0 , 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [9/91, .9890109890e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [57/637, .8948194662e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 3/7, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/13, .7692307692e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [17/234, .7264957265e-1, [7/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 8/9, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [32/455, .7032967033e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [135/2002, .6743256743e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/11, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [5/78, .6410256410e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/6, 1, 1]] -------------------------------------------- The deck has, 15, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [31/315, .9841269841e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0 , 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [23/210, .1095238095, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0 , 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [109/1050, .1038095238, [1, 4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [31/315, .9841269841e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [67/735, .9115646258e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/7, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [17/210, .8095238095e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [1/14, .7142857143e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/7, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [12/175, .6857142857e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [51/770, .6623376623e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/11, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [4/63, .6349206349e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1]] -------------------------------------------- The deck has, 16, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [71/720, .9861111111e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0 , 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [13/120, .1083333333, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0 , 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [127/1200, .1058333333, [1, 4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [7/72, .9722222222e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [11/120, .9166666667e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [79/1080, .7314814815e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/9, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [1/15, .6666666667e-1, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [57/880, .6477272727e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8/11, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [1/16, .6250000000e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1]] -------------------------------------------- The deck has, 17, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [27/272, .9926470588e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [15/136, .1102941176, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [29/272, .1066176471, [1, 4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [13/136, .9558823529e-1, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [87/952, .9138655462e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/7, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [23/272, .8455882353e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [31/408, .7598039216e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [9/136, .6617647059e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [189/2992, .6316844920e-1, [9/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10/11, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [25/408, .6127450980e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1]] -------------------------------------------- The deck has, 18, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [91/918, .9912854030e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [163/1530, .1065359477, [1, 4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [5/51, .9803921569e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [97/1071, .9056956116e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/7, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [13/153, .8496732026e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [107/1377, .7770515614e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 , 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/9, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [53/765, .6928104575e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [19/306, .6209150327e-1, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/9, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [55/918, .5991285403e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/6, 1, 1]] -------------------------------------------- The deck has, 19, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [17/171, .9941520468e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [181/1710, .1058479532, [1, 4/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 , 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [17/171, .9941520468e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [107/1197, .8939014202e-1, [1, 3/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [29/342, .8479532164e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [121/1539, .7862248213e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7/9, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [61/855, .7134502924e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [119/1881, .6326422116e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/11, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [10/171, .5847953216e-1, [5/6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- The deck has, 20, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1 ], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [21/190, .1105263158, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [2/19, .1052631579, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [17/190, .8947368421e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/ 5, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [8/95, .8421052632e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [3/38, .7894736842e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [69/950, .7263157895e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [137/2090, .6555023923e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/11, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [11/190, .5789473684e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1]] -------------------------------------------- The deck has, 21, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1 , 1], [0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [23/210, .1095238095, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [37/350, .1057142857, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 , 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [89/980, .9081632653e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/7, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [1/12, .8333333333e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [149/1890, .7883597884e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/9, 1, 1, 1, 1 ]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [11/150, .7333333333e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [31/462, .6709956710e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/11, 1, 1, 1] ] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [19/315, .6031746032e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/6, 1, 1, 1]] -------------------------------------------- The deck has, 22, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [23/231, .9956709957e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] ] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [17/154, .1103896104, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [41/385, .1064935065, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [23/231, .9956709957e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [9/98, .9183673469e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/7, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [19/231, .8225108225e-1, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [163/2079, .7840307840e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/9, 1, 1 , 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [17/231, .7359307359e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [173/2541, .6808343172e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7/11, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [43/693, .6204906205e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1]] -------------------------------------------- The deck has, 23, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [151/1518, .9947299078e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 , 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [28/253, .1106719368, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [27/253, .1067193676, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [25/253, .9881422925e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [327/3542, .9232072276e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/7, 1 , 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [21/253, .8300395257e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [59/759, .7773386034e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [93/1265, .7351778656e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [191/2783, .6863097377e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9/11, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [16/253, .6324110672e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/ 2, 1, 1, 1]] -------------------------------------------- The deck has, 24, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [55/552, .9963768116e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [61/552, .1105072464, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [49/460, .1065217391, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [9/92, .9782608696e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [17/184, .9239130435e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [31/368, .8423913043e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/ 4, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [191/2484, .7689210950e-1, [1, 5/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8/9, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [101/1380, .7318840580e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [19/276, .6884057971e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [53/828, .6400966184e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1]] -------------------------------------------- The deck has, 25, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [179/1800, .9944444444e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [11/100, .1100000000, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [53/500, .1060000000, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [89/900, .9888888889e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/ 3, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [129/1400, .9214285714e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/7, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [17/200, .8500000000e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [23/300, .7666666667e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/7, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [109/1500, .7266666667e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [227/3300, .6878787879e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/11, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [29/450, .6444444444e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/6, 1, 1, 1]] -------------------------------------------- The deck has, 26, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [97/975, .9948717949e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [36/325, .1107692308, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [69/650, .1061538462, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [97/975, .9948717949e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [417/4550, .9164835165e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/7, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [111/1300, .8538461538e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [151/1950, .7743589744e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/9, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [9/125, .7200000000e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [49/715, .6853146853e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/11, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [21/325, .6461538462e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]] -------------------------------------------- The deck has, 27, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [35/351, .9971509972e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [25/234, .1068376068, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/ 5, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [35/351, .9971509972e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [149/1638, .9096459096e-1, [1, 1, 1/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [10/117, .8547008547e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [55/702, .7834757835e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [25/351, .7122507122e-1, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [263/3861, .6811706812e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/11, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [68/1053, .6457739791e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/6, 1, 1, 1, 1]] -------------------------------------------- The deck has, 28, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [113/1134, .9964726631e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [3/28, .1071428571, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [113/1134, .9964726631e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [241/2646, .9108087680e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/7, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [43/504, .8531746032e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [179/2268, .7892416226e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/9, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [1/14, .7142857143e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [281/4158, .6758056758e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8/11, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [73/1134, .6437389771e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1]] -------------------------------------------- The deck has, 29, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [81/812, .9975369458e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [45/406, .1108374384, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [3/28, .1071428571, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [121/1218, .9934318555e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [9/98, .9183673469e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/7, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [69/812, .8497536946e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [193/2436, .7922824302e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7/9, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [21/290, .7241379310e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [299/4466, .6695029109e-1, [1, 7/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10/11, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [13/203, .6403940887e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1]] -------------------------------------------- The deck has, 30, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [16/145, .1103448276, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [31/290, .1068965517, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [43/435, .9885057471e-1, [1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [281/3045, .9228243021e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/7, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [49/580, .8448275862e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [23/290, .7931034483e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [53/725, .7310344828e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [1/15, .6666666667e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 4/9, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [83/1305, .6360153257e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1]] -------------------------------------------- The deck has, 31, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [103/930, .1107526882, [1, 1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [33/310, .1064516129, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [46/465, .9892473118e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [43/465, .9247311828e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [13/155, .8387096774e-1, [1, 1, 1/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [221/2790, .7921146953e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/9, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [57/775, .7354838710e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [229/3410, .6715542522e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/11, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [88/1395, .6308243728e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/6, 1, 1, 1, 1]] -------------------------------------------- The deck has, 32, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [99/992, .9979838710e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [55/496, .1108870968, [1, 1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [33/310, .1064516129, [1, 1, 1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [37/372, .9946236559e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [321/3472, .9245391705e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/7, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [21/248, .8467741935e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [235/2976, .7896505376e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/9, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [183/2480, .7379032258e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [741/10912, .6790689150e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/11, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [1/16, .6250000000e-1, [1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 33, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [79/792, .9974747475e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [39/352, .1107954545, [1, 1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [47/440, .1068181818, [1, 1, 1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [79/792, .9974747475e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [31/336, .9226190476e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/7, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [15/176, .8522727273e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [83/1056, .7859848485e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [13/176, .7386363636e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [265/3872, .6844008264e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/11, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [1/16, .6250000000e-1, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 34, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [56/561, .9982174688e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [62/561, .1105169340, [1, 1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [20/187, .1069518717, [1, 1, 1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [56/561, .9982174688e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [361/3927, .9192768016e-1, [1, 1, 6/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1, 1, 1, 1, 1, 1] ] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [16/187, .8556149733e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [263/3366, .7813428402e-1, [1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8/9, 1, 1, 1, 1, 1, 1] ] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [69/935, .7379679144e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [283/4114, .6878949927e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7/11, 1, 1, 1, 1, 1 ]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [71/1122, .6327985740e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/6, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 35, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [178/1785, .9971988796e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [66/595, .1109243697, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [318/2975, .1068907563, [1, 1, 1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [178/1785, .9971988796e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [109/1190, .9159663866e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1 , 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [3/35, .8571428571e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [93/1190, .7815126050e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [219/2975, .7361344538e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [129/1870, .6898395722e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9/11, 1, 1, 1 , 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [38/595, .6386554622e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 36, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [377/3780, .9973544974e-1, [1, 1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [8/75, .1066666667, [1, 1, 1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [94/945, .9947089947e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [809/8820, .9172335601e-1, [1, 1, 1, 4/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/7, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [3/35, .8571428571e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [223/2835, .7865961199e-1, [1, 1, 1, 1/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/9, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [11/150, .7333333333e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [29/420, .6904761905e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 , 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [9/140, .6428571429e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 37, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [133/1332, .9984984985e-1, [1, 1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [1/9, .1111111111, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [71/666, .1066066066, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [11/111, .9909909910e-1, [1, 1, 1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [859/9324, .9212784213e-1, [1, 1, 1, 4/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/7, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [19/222, .8558558559e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [79/999, .7907907908e-1, [1, 1, 1, 1/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [27/370, .7297297297e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4/5, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [337/4884, .6900081900e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/11 , 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [43/666, .6456456456e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 38, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [421/4218, .9981033665e-1, [1, 1, 1, 2/3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [78/703, .1109530583, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [751/7030, .1068278805, [1, 1, 1, 1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [70/703, .9957325747e-1, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [909/9842, .9235927657e-1, [1, 1, 1, 4/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5/7, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [60/703, .8534850640e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [502/6327, .7934250040e-1, [1, 1, 1, 1/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5 /9, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [51/703, .7254623044e-1, [1, 1, 2/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 , 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [1065/15466, .6886072676e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 4/11, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [91/1406, .6472261735e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 5/6, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 39, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [74/741, .9986504723e-1, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [82/741, .1106612686, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [61/570, .1070175439, [1, 1, 1, 1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [74/741, .9986504723e-1, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [137/1482, .9244264507e-1, [1, 1, 1, 4/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 , 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [21/247, .8502024291e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 , 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [530/6669, .7947218474e-1, [1, 1, 1, 1/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 7/9, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [18/247, .7287449393e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [373/5434, .6864188443e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 6/11, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [16/247, .6477732794e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 1, 1, 1, 1, 1, 1]] -------------------------------------------- The deck has, 40, cards -------------------------------------------- For bet size , 1, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 2, we have that [V,evalf(V), P1,P2] are [173/1560, .1108974359, [1, 1, 1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 3, we have that [V,evalf(V), P1,P2] are [167/1560, .1070512821, [1, 1, 1, 1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 4, we have that [V,evalf(V), P1,P2] are [1/10, .1000000000, [1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1 , 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 5, we have that [V,evalf(V), P1,P2] are [1009/10920, .9239926740e-1, [1, 1, 1, 4/7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 2/7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 6, we have that [V,evalf(V), P1,P2] are [11/130, .8461538462e-1, [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 1, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 7, we have that [V,evalf(V), P1,P2] are [31/390, .7948717949e-1, [1, 1, 1, 1/9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 1, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 8, we have that [V,evalf(V), P1,P2] are [11/150, .7333333333e-1, [1, 1, 1, 1/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 1/5, 1, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 9, we have that [V,evalf(V), P1,P2] are [391/5720, .6835664336e-1, [1, 1, 5/11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 8/11, 1, 1, 1, 1, 1, 1]] -------------------------------------------- For bet size , 10, we have that [V,evalf(V), P1,P2] are [101/1560, .6474358974e-1, [1, 1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 , 0, 0, 0, 0, 1/6, 1, 1, 1, 1, 1, 1]] ---------------------------- This ends this paper that took, 1176.954, seconds to generate.