CombinatoricsDifficulty 5.7AIME, harderProve itUnited States
Problem:
Sandra the Maverick has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do?
Solution
Solution:
Answer: 95
On any given day, there is a 91 chance that the second shoe that Sandra chooses makes a pair with the first shoe she chose. Thus the average number of problems she does in a day is 91, so, by the linearity of expectation, she does 95 problems total, on average.
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Source: MathNet,
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