jmc

Question

A bin has 5 white balls and k black balls in it, where k is an unknown positive integer. A ball is drawn at random from the bin. If a white ball is drawn, the player wins 1 dollar, but if a black ball is drawn, the player loses 1 dollar. If the expected loss for playing the game is 50 cents, then what is k?

Accepted Answer

There are 5 white balls and 5+k total balls, so the probability that a white ball is drawn is 55+k. Similarly, the probability that a black ball is drawn is k5+k. So E = 55+k(1) + k5+k(-1) = -12. Multiply both sides of the equation by 2(5+k) to get 10 - 2k = -5 -k, and we see that k = 15.