Collecting three prizes with unequal odds

Question

A machine dispenses one prize per draw. The three prize types are NOT equally likely: type A appears with probability $\frac{1}{2}$ , type B with $\frac{1}{3}$ , and type C with $\frac{1}{6}$ , independently each draw.

On average, how many draws until you have collected at least one of all three types?

Accepted Answer

The equal-odds shortcut (nHn) does not apply here, so we build the answer from scratch. Let T be the draw on which the collection finally completes. A clean way to get its average is the identity E[T] = \sum{t\ge 0} P(T t): the collection is still unfinished after t draws whenever at least one type is still missing. Probability a type is missing after t draws. Type i (with probability pi) is absent from all t draws with probability (1-pi)^t. By inclusion-exclusion, the chance that some type is missing after t draws is P(Tt) = \sumi (1-pi)^t - \sum{i<j}(1-pi-pj)^t + (1-p1-p2-p3)^t. Sum over t.…

Collecting three prizes with unequal odds

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