To double or not to double, that is the question
You are playing backgammon for a stake of $1. You have reached a mid-game position where you estimate you have a 1/3 chance of winning a "gammon" (thus winning double the stake, or $2, from your opponent), a 1/3 chance of winning a regular game (winning $1), and a 1/3 chance of losing (losing the $1 to your opponent.) Assume your opponent agrees with your estimate of the position.
You have an opportunity to "double". If you do and your opponent accepts, the stake goes up to $2 (and thus, winning a gammon would give you $4, and winning a regular game would give the winner $2, from the other player.) If your opponent refuses, the game is over and he pays you $1.
Should you double? If so, should your opponent accept?
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1 solution
As the position stands, you have a 1/3 chance of winning $2, a 1/3 chance of winning $1, and a 1/3 chance of losing $1. The last two cancel out, so your mathematical expectation is $2/3 (1/3 * 2, for the gammon.)
If you double, everything gets multiplied by two, so if your opponent accepted, your mathematical expectation would be $2/3 * 2 = $4/3. If he refuses, you for certain win $1, which is less than 4/3.
So you should double, and your opponent should refuse.