aez-notes

Chuck-a-Luck is a gambling game often played at carnivals and gambling houses. A player may bet on any one of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If the player's number appears one, two, or three of the dice, he receives respectively one, two, or three times his original stake plus his own money back; otherwise he loses his stake. What is the player's expected loss per unit stake? (Actually the player may distribute stakes on several numbers, but each such stake can be regarded as a separate bet.)

This is just a calculation. Start by observing that the number of dice that show the chosen number follows a binomial distribution. Then, compute the expected payout given that distribution of outcomes. Rather than defining the binomial distribution we can load the distrib package which has functionality for working with common distributions.

load(distrib)$
f(x) := pdf_binomial(x,3,1/6);
payout(x) := if is(equal(x, 0)) then (-1) else x;
expected_payout : sum(f(x) * payout(x), x, 0, 3);
print(
  expected_payout,
  "which is ",
  float(expected_payout)
);

So playing this game will cost you \(0.079\) units on average.