Two-sided collection with a biased coin

Question

A source emits one of 2 types per draw: type A with probability 0.7 and type B with probability 0.3, independently. On average, how many draws until you have seen both types at least once?

Accepted Answer

The first draw always gives some type, so the only question is how long you then wait for the other type. - With probability 0.7 the first draw is A; then you wait for B, which arrives on average \frac{1}{0.3} draws later. - With probability 0.3 the first draw is B; then you wait for A, on average \frac{1}{0.7} draws later. E[T] = 1 + 0.7\cdot\frac{1}{0.3} + 0.3\cdot\frac{1}{0.7} = 1 + \frac{7}{3} + \frac{3}{7} = \frac{79}{21} \approx 3.76. The same answer follows from the general weighted formula for two types, E[T] = \frac{1}{pA}+\frac{1}{pB}-\frac{1}{pA+pB} = \frac{10}{7}+\frac{10}{3}-1 = …

Two-sided collection with a biased coin

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