An urn contains k black balls and a single red ball. Peter and Paula draw withou

Question

An urn contains k black balls and a single red ball. Peter and Paula draw
without replacement balls from this urn, alternating after each draw until the
red ball is drawn. The game is won by the player who happens to draw the
single red ball. Peter is a gentleman and offers Paula the choice of whether
she wants to start or not. Paula has a hunch that she might be better off if
she starts; after all, she might succeed in the first draw. On the other hand,
if her first draw yields a black ball, then Peter’s chances to draw the red ball
in his first draw are increased, because then one black ball is already removed
from the urn. How should Paula decide in order to maximize her probability
of winning?

Explanation / Answer

The best strategy depends on the parity of k. If k is even then the player who starts drawing has an advantage, but if k is odd, Peter’s and Paula’s chances are equal. To see why, consider a slightly revised, but functionally equivalent procedure to play this game: first Peter and Paula draw balls as before, but without inspecting their color, continuing until the urn is empty. Each player then inspects the balls in front of him or her, and the player who finds the red ball in his or her own sample has won. Notice that with this revised procedure the winner will always be the same as with the original procedure. Therefore, the probability of winning for each player will also be the same as in the original game. Consider first the case that k is odd, so that the total number of balls, 1+k, is even. Thus, after all balls are removed, both players will have drawn an equal number of balls, namely 1+k 2 . Now, each half of the 1+k balls has the same probability to contain the red one. Thus, the probability for Peter and Paula to find the red ball contained in his or her own sample is 1 2 . However, when k is even, then the total number 1 + k of balls in the urn is odd. Thus, the player who starts will eventually have sampled 1 + k 2 balls, as opposed to only k 2 balls drawn by the other player. Thus, the ratio favoring the starting player is 1 + k 2 1 + k 2 + k 2 = 2 + k 2 + 2k For example, if k = 0, the player who starts will necessarily always win. If k = 2, the starting player collects a total of two balls out of three, and so has a probability of 2 3 to win. Therefore, Paula should generally start drawing: if k is even, she gains a real advantage, and if k is odd at least nothing is lost.

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