--------------------------------------- Statistical Analysis of a Count Your Chickens! Board Game By Shalosh B. Ekhad Consider the "Cooperative" ( i.e. Solitaire) game where one spins a spinner \ with, 3, possible outcomes labelled [Sheep, Cow, Fox] and there is board with , 9, squares that looks as follows [ , , Sheep, Cow, , Cow, , Sheep, {Cow, Sheep}] where the last square contains all the animals There is also a set of locations marked blue. These are in the following loc\ ations {3, 6} You start at the first square, and advance along the board, at the same time\ gathering chicks, according to the rules. Your goal is to arrive at the coop located at the last square with a total of, 8, chicks. The rules are as follows. If you get a Fox, then you remove one chick from t\ he coop (unless it is emtpy, then you do nothing) and you stay at the same place. Otherwise you locate the next occurrence, on\ the board of the animal that you spun, and move the pawn (chicken) to that location. You add the number of squares thus adva\ nced to the coop. If the new location is a blue square, then you add an additional chick to th\ e coop. You keep going until you reach the last square (where the coop is) If the number of chicks in the coop is, 8, then you won, otherwise you lost. Here we will analyze this game limiting the game to <=, 100, rounds. Note that the probability that the game will last more than, 100, rounds is , -41 0.85885119585990322775 10 Hence this is a very good approximation to the ideal case when there is no l\ imit to the number of rounds. The probabiliy of winning (under the above assumption) is 0.62757201646090534979 The expected number of chicks in the coop at the end is 7.3410853697776422971 The variance of the random variable "number of chicks at the end" is 1.1299371680917704094 The skewness (aka scaled 3rd moment) of the random variable "number of chick\ s at the end" is -1.9328878518876689155 The kurtosis (aka scaled 4th moment) of the random variable "number of chic\ ks at the end" is 7.1489921414181470166 The scaled , 5, -th moment about the mean is this random variable is -27.716631240371473503 The scaled , 6, -th moment about the mean is this random variable is 121.80522697877057968 The expected number of rounds until the end is 5.1562500000000000000 The variance of the random variable "number of rounds" is 3.4130859375000000000 The skewness (aka scaled 3rd moment) of the random variable "number of round\ s " is 1.1457306413128666120 The kurtosis (aka scaled 4th moment) of the random variable "number of roun\ ds" is 4.7736656290101739456 The scaled , 5, -th moment about the mean is this random variable is 15.009361395748461234 The scaled , 6, -th moment about the mean is this random variable is 63.506013822612907608 The correlation between the number of chicks and the number of rounds is -0.66922985041532405161 Finally, the bi-variate probability generating function in the variables , X, c d t, whose coeficient of , X t , is the probability that the game ends after, c, rounds and with , d, chicks , is -10 We ignore terms smaller than , 0.50000000000000000000 10 -10 25 -9 24 -9 23 (0.7671530017 10 X + 0.2301459005 10 X + 0.6904377015 10 X -8 22 -8 21 -7 20 + 0.2071313104 10 X + 0.6213939313 10 X + 0.1864181794 10 X -7 19 -6 18 -6 17 + 0.5592545382 10 X + 0.1677763615 10 X + 0.5033290844 10 X -5 16 -5 15 14 + 0.1509987253 10 X + 0.4529961759 10 X + 0.00001358988528 X 13 12 11 + 0.00004076965584 X + 0.0001223089675 X + 0.0003669269025 X 10 9 8 + 0.001100780708 X + 0.003302342123 X + 0.009907026368 X 7 6 5 4 + 0.02972107910 X + 0.06447187929 X + 0.1358024691 X + 0.1975308642 X 3 8 -10 26 -9 25 + 0.1851851852 X ) t + (0.5586447499 10 X + 0.1640527188 10 X -9 24 -8 23 -8 22 + 0.4815360380 10 X + 0.1412741758 10 X + 0.4142626209 10 X -7 21 -7 20 -6 19 + 0.1214108143 10 X + 0.3556285269 10 X + 0.1041073833 10 X -6 18 -6 17 -5 16 + 0.3045786254 10 X + 0.8905053031 10 X + 0.2601824190 10 X -5 15 14 13 + 0.7596397412 10 X + 0.00002216196676 X + 0.00006460422386 X 12 11 10 + 0.0001881676423 X + 0.0005475678391 X + 0.001591898254 X 9 8 7 + 0.004623278972 X + 0.01341258954 X + 0.02240512117 X 6 5 4 7 + 0.05075445816 X + 0.05349794239 X + 0.04938271605 X ) t + ( -10 28 -9 27 -9 26 0.5280460547 10 X + 0.1476605607 10 X + 0.4119021501 10 X -8 25 -8 24 -8 23 + 0.1146008561 10 X + 0.3179554133 10 X + 0.8795114105 10 X -7 22 -7 21 -6 20 + 0.2425029650 10 X + 0.6663254925 10 X + 0.1824030186 10 X -6 19 -5 18 -5 17 + 0.4973063432 10 X + 0.1349954416 10 X + 0.3647199981 10 X -5 16 15 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0.2410302935 10 X + 0.6396875798 10 X + 0.1688523427 10 X -8 25 -7 24 -7 23 + 0.4430603643 10 X + 0.1154978350 10 X + 0.2989064142 10 X -7 22 -6 21 -6 20 + 0.7673418393 10 X + 0.1952132935 10 X + 0.4915703992 10 X -5 19 -5 18 -5 17 + 0.1223476851 10 X + 0.3004487458 10 X + 0.7263425864 10 X 16 15 14 + 0.00001723708526 X + 0.00004000304692 X + 0.00009032046831 X 13 12 11 + 0.0001969487990 X + 0.0002408545822 X + 0.0005306327513 X 10 9 2 -10 31 + 0.0003048315806 X + 0.0004064421074 X ) t + (0.8039523848 10 X -9 30 -9 29 -8 28 + 0.2074591536 10 X + 0.5320093143 10 X + 0.1354997649 10 X -8 27 -8 26 -7 25 + 0.3425305368 10 X + 0.8587392677 10 X + 0.2133157439 10 X -7 24 -6 23 -6 22 + 0.5244493966 10 X + 0.1274441776 10 X + 0.3055983488 10 X -6 21 -5 20 -5 19 + 0.7215817529 10 X + 0.1673174859 10 X + 0.3796047727 10 X -5 18 17 16 + 0.8383655723 10 X + 0.00001788754131 X + 0.00003642553866 X 15 14 13 + 0.00006927356906 X + 0.0001175002389 X + 0.0001555519176 X 12 11 10 + 0.0002258011708 X + 0.0001467707610 X + 0.0001354807025 X ) t and in Maple notation (.7671530017e-10*X^25+.2301459005e-9*X^24+.6904377015e-9*X^23+.2071313104e-8*X^ 22+.6213939313e-8*X^21+.1864181794e-7*X^20+.5592545382e-7*X^19+.1677763615e-6*X ^18+.5033290844e-6*X^17+.1509987253e-5*X^16+.4529961759e-5*X^15+.1358988528e-4* X^14+.4076965584e-4*X^13+.1223089675e-3*X^12+.3669269025e-3*X^11+.1100780708e-2 *X^10+.3302342123e-2*X^9+.9907026368e-2*X^8+.2972107910e-1*X^7+.6447187929e-1*X ^6+.1358024691*X^5+.1975308642*X^4+.1851851852*X^3)*t^8+(.5586447499e-10*X^26+.\ 1640527188e-9*X^25+.4815360380e-9*X^24+.1412741758e-8*X^23+.4142626209e-8*X^22+ .1214108143e-7*X^21+.3556285269e-7*X^20+.1041073833e-6*X^19+.3045786254e-6*X^18 +.8905053031e-6*X^17+.2601824190e-5*X^16+.7596397412e-5*X^15+.2216196676e-4*X^ 14+.6460422386e-4*X^13+.1881676423e-3*X^12+.5475678391e-3*X^11+.1591898254e-2*X ^10+.4623278972e-2*X^9+.1341258954e-1*X^8+.2240512117e-1*X^7+.5075445816e-1*X^6 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