Q: One urn contains black marbles, and the other contains white or black marbles with even odds. You pick a marble from an urn; it is black; you put it back; what are the odds that you will draw a black marble on the next draw? What are the odds after n black draws?

A: Every time you draw a black marble, you throw out (from your probability space) half of those possible urns that contain both colors. So you have 1/2^n times as many ways to have a white marble in the urn after n draws, all black, as at the start. But you have exactly the same number of ways to have both marbles black. The numbers (mixed cases vs. all-black cases) go as 1:1, 1:2, 1:4, 1:8,... and the chance of having a white marble in the urn goes as 1/2, 1/3, 1/5, 1/9, ..., 1/(1+2^(n-1)), hence the odds of drawing a white marble on the nth try after n-1 consecutive drawings of black are

1/4 the first time 1/6 the second time 1/10 the third time ... 1/(2+2^n) the nth time

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