How many rolls to see all three colors?

Question

A spinner has 3 equally likely colors: red, green, and blue. You spin repeatedly, independently, until all 3 colors have shown up at least once. On average, how many spins does that take?

Accepted Answer

Split into 3 stages by how many distinct colors you have seen, and add the stage lengths with linearity of expectation. The first spin is always a new color, so stage 1 takes exactly 1 spin. With 1 color seen, a new color comes with probability 	frac23, taking on average 	frac32 spins. With 2 colors seen, the last color comes with probability 	frac13, taking on average 3 spins. Summing, E[	ext{spins}] = 1 + \frac32 + 3 = 3\left(\frac11 + \frac12 + \frac13ight) = 3 	imes \frac{11}{6} = \frac{11}{2} = 5.5. Takeaway. For 3 equally likely outcomes the collection wait is 3\left(1 + 	frac12…

How many rolls to see all three colors?

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