Question

A box contains 4 white balls and 4 black balls, and the following game is played. Each round of the game consists of randomly selecting four balls from the box. If exactly two of them are white, the game is over. Otherwise, the four balls are placed back in the box, so that it again contains 4 white balls and 4 black balls before the next round is performed. The rounds are repeated until exactly two of the chosen balls are white. (a) Let X denote the number of rounds in the game. Find the expected value and variance of X. (b) When the game is over, we get a reward Y. The value of Y decreases exponentially with the the number of rounds played. Specifically, if the game terminates after n rounds, then Y = 1/e^n. Find the expected value of the reward Y.

Answer 1