Prize types among four spins

Question

A prize wheel has 3 equally likely prize types. You spin it 4 times, each spin independent. On average, how many distinct prize types do your 4 spins cover?

Accepted Answer

Linearity of expectation: sum, over the 3 prize types, the probability each appears at least once. A specific prize is missed on a single spin with probability \frac{2}{3}, so it is absent from all 4 spins with probability \left(\frac{2}{3}ight)^4 = \frac{16}{81}. It appears at least once with probability 1 - \frac{16}{81} = \frac{65}{81}. With 3 prize types, E[	ext{distinct}] = 3\left(1 - \left(	frac23ight)^4ight) = 3 	imes \frac{65}{81} = \frac{65}{27} \approx 2.41. Takeaway. Expected distinct types after m draws is n\left(1 - \left(\frac{n-1}{n}ight)^might); four spins cover a…

Prize types among four spins

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