Keep it or draw again

EasyProbability Expected Value~4m

You draw a number uniformly from $(0,1)$ . You may keep it, or discard it and draw once more - in which case your payout is the second draw (you cannot go back). Playing optimally, what is your expected payout?

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A fresh draw is worth $1/2$ on average, so keep the first draw only if it exceeds $1/2$ .
Combine: (keep) $\mathbb{P}(X\ge\tfrac12)\,\mathbb{E}[X\mid X\ge\tfrac12]$ plus (redraw) $\mathbb{P}(X<\tfrac12)\cdot\tfrac12$ .

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0.625

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Asked at: Jane Street, Optiver

Keep it or draw again

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