Distinct cards from six single-card packs

Question

There are 10 different football cards, and each pack contains one card that is equally likely to be any of the 10, independently across packs. James opens 6 packs. What is the expected number of different cards he ends up with?

Short answer

$10\left(1-0.9^6\right)\approx 4.69$ . Sum one indicator per card type: a given card is missing from all six packs with probability $0.9^6$ , so by linearity the expected number of distinct cards is $10(1-0.9^6)$ .

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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