How many coupon types after four purchases?

Question

A cereal brand includes one coupon in each box, drawn uniformly at random from 5 possible coupon types. You buy 4 boxes. On average, how many distinct coupon types do you end up with?

Accepted Answer

Trying to sum over all the ways duplicates can occur is painful. Put one indicator on each coupon type instead. Fix a coupon type. You fail to collect it only if all 4 boxes contain one of the other types. Each box avoids a given type with probability 	frac45, so P(	ext{a given type is missing}) = \left(	frac45ight)^4 = \frac{256}{625}. Then you own a given type with probability 1 - 	frac{256}{625} = 	frac{369}{625}. Let Xi = 1 if you have type i; E[Xi] = 	frac{369}{625}. With 5 types, E[X] = 5 	imes \frac{369}{625} = \frac{369}{125} = 2.952. Takeaway. The number of distinct items co…

How many coupon types after four purchases?

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