Three coupons in a snack promotion

Question

A snack promotion prints one coupon inside each bag. There are 3 coupon types, each equally likely and independent between bags.

On average, how many bags must you open to collect all three coupon types?

Accepted Answer

Sort your progress by how many different coupons you hold. The very first bag always gives a brand-new coupon, so stage one costs 1 bag on average. Once you have 1 type, a bag is new with probability \frac{2}{3}, so it takes on average \frac{3}{2} bags to reach 2 types. Once you have 2 types, only \frac{1}{3} of bags help, so the last type takes on average \frac{3}{1} = 3 bags. E = \frac{3}{3} + \frac{3}{2} + \frac{3}{1} = 1 + 1.5 + 3 = 5.5 	ext{ bags.} Takeaway. With n equally likely types the average is n Hn; for three types, 3H3 = \frac{11}{2} = 5.5.

Three coupons in a snack promotion

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