- #1
xcrunner448
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Homework Statement
A cereal box company started a promotion in which they put one of five different baseball cards in each box of cereal. All cards occur with equal probability. What is the expected number of cereal boxes you have to buy before you collect all five cards?
The Attempt at a Solution
The expected value is the sum from k=5 to infinity of k*(probability of taking exactly k boxes to collect all 5 cards). However, I cannot find a general expression that gives that probability. For k=5, it would be (0.2^5)*5! = 0.0384, because the probability of a certain card in each box is 0.2, and there are 5!=120 different ways to collect all five cards. I cannot figure out how to calculate this probability for k>5. I have tried the same method as I used for k=5, that is, (0.2^k) times the number of ways of collecting all five cards, but because some cards must be repeated 1 or more times I am having difficulty finding out how to calculate that.