Halfway through a set of twelve

Question

A stamp series has 12 designs, each equally likely and independent per packet. You have collected 6 different designs so far. On average, how many more stamps must you obtain to complete all twelve?

Accepted Answer

Only the 6 missing designs matter now. Progress in stages by the number of distinct designs held, from 6 up to 12. With k designs, a new one arrives with probability \frac{12-k}{12}, so the average wait is \frac{12}{12-k} stamps. Sum over k = 6,7,8,9,10,11: E = \frac{12}{6}+\frac{12}{5}+\frac{12}{4}+\frac{12}{3}+\frac{12}{2}+\frac{12}{1} = 12\left(	frac16+	frac15+	frac14+	frac13+	frac12+1ight). The bracket is 2.45, so E = 12 	imes 2.45 = 29.4 	ext{ more stamps.} Even though you are halfway by count, the remaining half takes far longer than the first, since the useful fraction keeps s…

Halfway through a set of twelve

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