Expected spins to reveal all five prizes

Question

A prize wheel shows one of five distinct prizes each spin, each equally likely and independent. You spin until all five prizes have shown up. What is the expected number of spins?

Accepted Answer

With j prizes already seen, a spin reveals a new one with probability \frac{5 - j}{5}, and the expected spins to advance is ej = 1 + \frac{j}{5}ej = \frac{5}{5 - j}. Summing over j = 0,\dots,4: E = \frac55 + \frac54 + \frac53 + \frac52 + \frac51 = 5\Big(1 + 	frac12 + 	frac13 + 	frac14 + 	frac15\Big) = 5 \cdot \frac{137}{60} = \frac{137}{12} \approx 11.42. Takeaway. Five equally likely prizes take 5 \cdot H5 = 	frac{137}{12} spins on average; the harmonic sum is the whole story.

Expected spins to reveal all five prizes

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