Distinct toys from seven boxes

HardProbability Expected Value~7m

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

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Linearity over the $5$ toys: $E[\text{distinct}]=5\cdot\mathbb{P}[\text{a given toy appears}]$ .
A toy is missing with probability $(4/5)^7$ , so $E=5(1-(4/5)^7)$ .

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3.9514 (± 0.01)

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Asked at: Two Sigma, Akuna Capital

Distinct toys from seven boxes

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