Expected draws to collect all ten coupons

Question

A promotion has 10 distinct coupons, each equally likely in every draw and independent between draws. What is the expected number of draws to collect all 10? Give a number and the rule of thumb for large n.

Accepted Answer

With n = 10 equally likely coupons, split the wait into stages T = T1 + \cdots + T{10}, where Tk is the draws for the k-th new coupon given k - 1 held. Holding k - 1, a draw is new with probability pk = 	frac{10 - (k-1)}{10}, so Tk is geometric with mean 	frac{10}{11 - k}. Summing, E[T] = 10\left(1 + \frac12 + \frac13 + \cdots + \frac{1}{10}ight) = 10\,H{10}. Using H{10} = 	frac{7381}{2520} \approx 2.9290, E[T] = 10 	imes 2.9290 \approx 29.29 	ext{ draws.} Rule of thumb. For large n, Hn \approx \ln n + \gamma with \gamma \approx 0.5772 (the Euler-Mascheroni constant), so E[T] \approx n…

Expected draws to collect all ten coupons

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