Keep a drawn card or redraw once, played optimally

Question

You draw a card whose value is equally likely to be any whole number from $1$ to $10$ . You may keep it, or redraw once and must take the second card.

Playing to maximize your expected value, what value should you expect?

Accepted Answer

Fix the rule, then condition on the first card (the tower rule). Step 1, the redraw's worth. A fresh card averages E[	ext{redraw}] = \frac{1 + 10}{2} = 5.5. So keep the first card only when it beats 5.5, that is when it is 6 through 10, and redraw on 1 through 5. Step 2, condition on the first card. - First card is 6 through 10 (probability 	frac12): keep it, averaging \frac{6 + 10}{2} = 8. - First card is 1 through 5 (probability 	frac12): redraw, averaging 5.5. Step 3, combine: E[	ext{value}] = 	frac12 	imes 8 + 	frac12 	imes 5.5 = 4 + 2.75 = 6.75. Your expected value under the best…

Keep a drawn card or redraw once, played optimally

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