Computes the estimation of expected gain for a game wherein the following rules apply: A player pays p_0 to play. The player then rolls a 6-sided die. If the roll is a 1 or a 2 then the game is over. Otherwise, the player flips a fair coin. If the coin toss results in a tail, they receive $1 and the game is over. Otherwise they draw 2 cards without replacement from the standard deck of 52 cards. If none of the cards is an ace they receive $2, while they receive $10 or $50 is they get 1 or 2 aces respectively. In both cases, the game is over. Let G denote the player's gain. To determine the expected gain, we need the distribution of G. The support of G is the set -p_0, 1-p_0, 2-p_0, 10-p_0, 50-p_0. For the associated probabilities we need the distribution of X where X is the number of aces in a draw of 2 cards. The distribution is P(X=x) = ( (4 choose x) * (48 choose (2-x)) ) / (52 choose 2) for x=0,1,2
See example 1.8.9 on page 65 of the book.
Initial amount that a player pays (p_0 in description).
Estimation of expected gain for player.
Hogg, R. McKean, J. Craig, A (2018) Introduction to Mathematical Statistics, 8th Ed. Boston: Pearson
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