Two numbers, keep the larger

Question

Two numbers are drawn independently and uniformly on $[0, 1]$ . The first is shown to you; you may keep it (ending the game) or pass to the second, which you must then keep. You win if the number you keep is the larger of the two.

What threshold policy maximizes your win probability, and what is that probability?

Accepted Answer

Let the first draw be x and the second be y, both uniform on [0,1]. Use the policy: keep x if x t, otherwise pass to y. - If you keep x (when x t), you win if x y, which given x happens with probability x. - If you pass (when x \le t), you keep y and win if y x, which given x happens with probability 1 - x. Integrate over the uniform first draw: P(	ext{win}) = \intt^1 x\,dx + \int0^t (1 - x)\,dx = \left(	frac12 - 	frac{t^2}{2}ight) + \left(t - 	frac{t^2}{2}ight) = 	frac12 + t - t^2. Maximize: \dfrac{d}{dt}\left(	frac12 + t - t^2ight) = 1 - 2t = 0 \implies t = 	frac12. The win pro…

Two numbers, keep the larger

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