Spins to land on all three prize colors

Question

A prize wheel has 3 equal-size colors: red, green, and blue. Each spin lands on a color uniformly at random and independently. You spin until you have landed on all 3 colors at least once. On average, how many spins does that take?

Accepted Answer

Coupon collecting with 3 equally likely types. With k colors already seen, a spin gives a new color with probability \frac{3 - k}{3}, averaging \frac{3}{3 - k} spins for that stage. Summing k = 0, 1, 2: E = \frac{3}{3} + \frac{3}{2} + \frac{3}{1} = 3\left(1 + 	frac12 + 	frac13ight) = 3 \cdot \frac{11}{6} = \frac{11}{2} = 5.5. Takeaway. Collecting all 3 equally likely colors averages 3 H3 = 5.5 spins.

Spins to land on all three prize colors

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