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Yes. Conceptually, color the interval [0,p[ as HEADS and the interval [p,1[ as TAILS, where we use [a,b[ to mean "all numbers from a, inclusive, up through b, exclusive".

Now, given an interval with two colors, divide it evenly in half. Use the fair coin to randomly pick one half. If the chosen half has only one color, that is the answer. Otherwise, use the new half as the interval, and repeat the process.

How many flips F should you expect to take? Well, at least half the time you stop immediately (end in one interval with just one color), and the other half, you need another F flips. So: F = 1 * 1/2 + (1+F) * 1/2. Multiply through by 2 and simplify to get 2 * F = 1 + 1 + F. Solving for F gives you 2 expected flips.

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