Keep your card or swap to beat a hidden one

Question

You and an opponent each secretly receive a number drawn uniformly at random from $1$ to $10$ (the two draws are independent). You get to look at yours. You may keep it, or swap it for one fresh uniform draw from $1$ to $10$ (and you must keep the new one). The opponent never changes their number. Whoever ends with the higher number wins; ties count as a loss for you.

What is your optimal swap rule, and what is your resulting probability of winning?

Accepted Answer

If you hold a number x, you win exactly when the opponent's hidden number is below x. Since their number is uniform on 1 to 10, P(	ext{win} \mid 	ext{keep } x) = \frac{x - 1}{10}. If instead you swap, your new number Y is uniform on 1 to 10, and your win chance averages out to P(	ext{win} \mid 	ext{swap}) = E\!\left[\frac{Y-1}{10}ight] = \frac{5.5 - 1}{10} = 0.45. So swap whenever keeping wins with probability below 0.45, that is when 	frac{x-1}{10} < 0.45, meaning x < 5.5. Keep 6 through 10, swap 1 through 5. Now combine. Half the time (a 1 to 5) you swap and win with chance 0.45. Hal…

Keep your card or swap to beat a hidden one

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