One optional redraw of a uniform number

Question

You draw a number uniformly from $(0,1)$ . You may either accept it or discard it and draw one fresh uniform number, which you must then accept. You are paid whichever number you end with. Playing optimally, what is your expected payout?

Short answer

$5/8$ . A redraw is worth $1/2$ on average, so keep the first number exactly when it exceeds $1/2$ . Then the payout is $\tfrac12\cdot\tfrac34+\tfrac12\cdot\tfrac12=\tfrac58$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Continuous Random Variables and Probability Density Functions
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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