Distinct prizes from seven cereal boxes

Question

Each cereal box contains one of $5$ different toys, chosen uniformly and independently from box to box. If you buy $7$ boxes, what is the expected number of different toys you end up with?

Short answer

$61741/15625\approx3.95$ . Use one indicator per toy: a given toy is missing from all seven boxes with probability $(4/5)^7$ . By linearity the expected count is $5\left(1-(4/5)^7\right)=\tfrac{61741}{15625}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Independence
Conditional Expectation and Variance Revisited

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