Keep a spin or respin once on the unit dial

Question

A dial spins to a number uniform on [0, 1]. You may keep it, or respin once and must take the second spin. Playing to maximize your expected value, what value should you expect?

Accepted Answer

Choose the rule, then condition on the first spin (the tower rule). The reasoning is the same as for a die, just with a continuous value. Step 1, the respin's worth. A fresh spin averages E[	ext{respin}] = 0.5. So keep the first spin only when it beats 0.5, and respin when it is below 0.5. Each case has probability 	frac12. Step 2, condition on the first spin. - First spin is above 0.5 (probability 	frac12): keep it. A uniform value given that it lies in [0.5, 1] averages the midpoint 0.75. - First spin is below 0.5 (probability 	frac12): respin, averaging 0.5. Step 3, combine: E[	ext{va…

Keep a spin or respin once on the unit dial

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