Statistical Analysis of a certain Chutes-and-Ladders Game



                              By Shalosh B. Ekhad



Consider the following game with a 1-dimensional board with, 30,

    squares numbered from  1 to, 30

You start at square 1 and proceed by rolling a die (or spinner) with the fol\
    lowing possible outcomes.



                     It lands on, 1, with probability , 1/2

                     It lands on, 2, with probability , 1/2

At each turn, you roll the die (or spin the spinner), and advance forward th\
    e number of squares indicated.

Your goal is to reach the last square, square, 30,  it does not matter if in \
    the last round you went beyond it, either way the game ended.



The probabilty generating function for the random variable, "number of turns\
     to get to the end"  is



 15   14       13         12          11           10            9            8
t   (t   + 55 t   + 1352 t   + 19600 t   + 186208 t   + 1218448 t  + 5617920 t

                 7             6             5             4             3
     + 18356736 t  + 42170880 t  + 66646528 t  + 69701632 t  + 45260800 t

                 2
     + 16400384 t  + 2723840 t + 131072)/268435456



                          and in Maple notation it is



1/268435456*t^15*(t^14+55*t^13+1352*t^12+19600*t^11+186208*t^10+1218448*t^9+
5617920*t^8+18356736*t^7+42170880*t^6+66646528*t^5+69701632*t^4+45260800*t^3+
16400384*t^2+2723840*t+131072)


                The expected duration of the (solitaire) game is



                                  19.55555556



            The variance of the duration of the (solitaire) game is



                                  2.172839498



                  The scaled , 3, -th moment about the mean is



                                  0.2124284834



                  The scaled , 4, -th moment about the mean is



                                  2.976755920



Summarizing the, expectation, variance and the scaled moments up to the, 4,

    -th , are



             [19.55555556, 2.172839498, 0.2124284834, 2.976755920]



Let's compare it to simulating , 1000, random games , the corresponding list is



             [19.62400000, 2.048624000, 0.2257709435, 3.162342845]



[Of course this changes (hopefully only slightly, by the law of large number\
    s)  each time]





So far the game was Solitaire. Suppose that there are two players, and they \
    keep taken turns.

           Assuming that the game did not last beyond, 1000,  moves



           The probability, that the first player is going to win is

                                  0.5955725767



Just for fun, let's compare it to the result of simulating, 1000,

    such games, and see the fraction of times the first player won



                                  0.5740000000

                           -------------------------



        This ends this article, that took , 2.542,  seconds to generate.