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Keep or redraw

Expected valueEasy

A machine shows a random integer from $1$ to $10$ , each equally likely. You may keep it and be paid that many dollars, or reject it and take one fresh independent draw, which you must keep. You play to maximise your expected payout. What is that expected payout?

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Hint

Compare each shown number with the average of a fresh draw.

Solution

A fresh draw is worth $\frac{11}{2}$ on average, so keep any number above $\frac{11}{2}$ , that is $6$ to $10$ .
Keeping happens with probability $\frac{1}{2}$ and the kept number averages $8$ ; otherwise you redraw and expect $\frac{11}{2}$ .
The expected payout is $\frac{1}{2} \times 8 + \frac{1}{2} \times \frac{11}{2} = \frac{27}{4}$ .

Answer: $\frac{27}{4}$

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