Three flavors left to find: how many more scoops?

Question

A shop serves 5 flavors, each equally likely on a random scoop, independently every time. You have already tasted 2 distinct flavors and want all 5.

On average, how many more scoops until you have collected the remaining 3 flavors?

Accepted Answer

You still need 3 new flavors, so there are 3 stages left. Add their lengths with linearity of expectation. Right now you own 2 flavors, so 3 of the 5 are new; the next new flavor arrives with probability 	frac35, taking on average 	frac53 scoops. After that, 2 of 5 are new, taking 	frac52 scoops on average. Finally, 1 of 5 is new, taking 	frac51 = 5 scoops on average. Summing, E[	ext{more scoops}] = \frac53 + \frac52 + 5 = 5\left(\frac13 + \frac12 + 1ight) = 5 	imes \frac{11}{6} = \frac{55}{6} \approx 9.167. Takeaway. Coupon collecting works from any starting point: only add the stage…

Three flavors left to find: how many more scoops?

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