Problem of the Week 1212

Self-Referential Probability

Alice has a coin that comes up HEADS with probability q; Bob has the same, with probability r. They alternate throws, with Alice going first. The first to get a HEAD wins.

Assume that q and r are random, independent, and uniformly distributed in [0, 1]. Let p(q, r) be the chance that, for the specified q and r, Alice wins.

Is there a value X such that X equals the probability (as q and r vary) that p(q, r) ≥ X?

Source: original problem of Steven J. Miller (Williams College) and Stan Wagon (Macalester College).

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