How many chips so every cookie gets one

Question

A baker mixes $N$ chocolate chips into dough for $100$ cookies, and each chip independently ends up in any of the $100$ cookies with equal probability. What is the smallest $N$ for which there is at least a $90\%$ chance that no cookie is chip-free?

Short answer

$N=683$ . By inclusion–exclusion, $\mathbb{P}(\text{all covered})=\sum_{j}(-1)^j\binom{100}{j}(1-\tfrac{j}{100})^N$ , which first reaches $0.9$ at $N=683$ . A quick estimate uses the expected number of empty cookies: $100(0.99)^N\approx0.1$ gives $N\approx\ln 1000/0.01\approx690$ .

Adapted from Crack, Heard on the Street, Q4.32, p.43; question and answer rewritten by SelfMadeQuant.com.

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