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, 6, possible outcomes labelled [Cow, Dog, Pig, Sheep, Tractor, Fox] and there is board with , 41, squares that looks as follows [ , , Sheep, Pig, Tractor, Cow, Dog, Pig, Cow, Dog, Sheep, Tractor, , Cow, Pig, , , , Tractor, , Tractor, Dog, Sheep, Cow, Dog, Pig, Tractor, , Sheep, Cow, , , Tractor, Pig, Sheep, Dog, , Sheep, Cow, Pig, {Cow, Dog, Pig, Sheep, Tractor}] where the last square contains all the animals There is also a set of locations marked blue. These are in the following loc\ ations {5, 9, 23, 36, 40} 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, 40, 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, 40, then you won, otherwise you lost. Here we will analyze this game limiting the game to <=, 80, rounds. Note that the probability that the game will last more than, 80, rounds is , -36 0.89185709281639359848 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 0.64103739962313655667 The expected number of chicks in the coop at the end is 39.322304391423431771 The variance of the random variable "number of chicks at the end" is 1.2907513179745173166 The skewness (aka scaled 3rd moment) of the random variable "number of chick\ s at the end" is -2.0548902227966114446 The kurtosis (aka scaled 4th moment) of the random variable "number of chic\ ks at the end" is 7.8590953453453550353 The scaled , 5, -th moment about the mean is this random variable is -32.804144889174185118 The scaled , 6, -th moment about the mean is this random variable is 160.05451848354814938 The expected number of rounds until the end is 11.447067101613353025 The variance of the random variable "number of rounds" is 6.2803011216806430055 The skewness (aka scaled 3rd moment) of the random variable "number of round\ s " is 0.56530276229872264081 The kurtosis (aka scaled 4th moment) of the random variable "number of roun\ ds" is 3.4188881018531040976 The scaled , 5, -th moment about the mean is this random variable is 6.0493018857079733301 The scaled , 6, -th moment about the mean is this random variable is 24.933323696907909142 The correlation between the number of chicks and the number of rounds is -0.52778542190743271167 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 29 -9 28 -8 27 (0.8162005567 10 X + 0.4721932760 10 X + 0.2679843071 10 X -7 26 -7 25 -6 24 + 0.1473300999 10 X + 0.7739644428 10 X + 0.3836028512 10 X -5 23 -5 22 21 + 0.1775163526 10 X + 0.7609435396 10 X + 0.00003004076268 X 20 19 18 + 0.0001087532827 X + 0.0003597730765 X + 0.001084051075 X 17 16 15 + 0.002964570900 X + 0.007326469086 X + 0.01627198785 X 14 13 12 + 0.03224031271 X + 0.05642046335 X + 0.08601930009 X 11 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