Expected boxes to collect all five toys

Question

A cereal brand puts one of 5 different toys in each box, each toy equally likely and independent between boxes. What is the expected number of boxes you must buy to collect all 5 toys?

Accepted Answer

This is the coupon collector problem with n = 5. Write the total as T = T1 + T2 + T3 + T4 + T5, where Tk is the number of boxes to get the k-th new toy once you already hold k - 1 distinct toys. While you hold k - 1 distinct toys, each box is new with probability pk = 	frac{5 - (k-1)}{5}, so Tk is geometric with mean 	frac{1}{pk} = 	frac{5}{6 - k}. By linearity of expectation, E[T] = \frac55 + \frac54 + \frac53 + \frac52 + \frac51 = 5\left(1 + \frac12 + \frac13 + \frac14 + \frac15ight) = 5\,H5. Since H5 = 1 + 	frac12 + 	frac13 + 	frac14 + 	frac15 = 	frac{137}{60}, E[T] = 5 \cdot \fr…

Expected boxes to collect all five toys

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