Question

You are a baseball card collector. For simplicity, let’s assume there are only seven kind of baseball cards. Each time you purchase a card, you have equal chance of getting each of the seven kinds. Calculate the expected value of total number of cards you need to purchase to collect all seven kinds of cards. (Hint: Treat it as geometric distributions with gradually decreasing success rates.)

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Answer 1

The time until the first result appears is 1. After that, the random time until a second (different) result appears is geometrically distributed with parameter of success 6/7, hence with mean 7/6(since the mean of a geometrically distributed random variable is the inverse of its parameter). After that, the random time until a third (different) result appears is geometrically distributed with parameter of success 5/7, hence with mean 7/5. And so on, until the random time of appearance of the last and seventh result, which is geometrically distributed with parameter of success 1/7, hence with mean 7/1. This shows that the mean total time to get all seven results is

1+7/6 + 7/5 + 7/4 + 7/3 + 7/2 + 7/1 = 18.15