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Question

A bin has 5 green balls and k purple balls in it, where k is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 2 dollars, but if a purple ball is drawn, the player loses 2 dollars. If the expected amount won for playing the game is 50 cents, then what is k?

Accepted Answer

There are 5 green balls and 5+k total balls, so the probability that a green ball is drawn is 55+k. Similarly, the probability that a purple ball is drawn is k5+k. So the expected value is 55+k(2)+k5+k(-2)=12. Multiplying both sides of the equation by 2(5+k) gives 20-4k=5+k, or 15=5k. Therefore, k=3.