Solution
The Dice Duel
Show the problem again
Two players each roll a fair six-sided die. The player with the lower number pays the other player the difference between the two numbers, in dollars (nothing changes hands on a tie). What is the expected amount of money that changes hands, that is, E|X − Y| where X and Y are the two rolls? Give a reduced fraction (a decimal rounded to two places is also accepted).
Worked solution
The answer is 35/18 dollars, about $1.94. Count the 36 equally likely outcomes by the difference d = |X − Y|. A difference of d ≥ 1 occurs in 2(6 − d) ordered outcomes, so the total of |X − Y| over all outcomes is 2(5·1 + 4·2 + 3·3 + 2·4 + 1·5) = 2 × 35 = 70. Dividing by 36 gives 70/36 = 35/18. The game itself is fair by symmetry (each player's expected net is 0); the question measures how much is at stake, not who wins it.
Source: Standard expected-value exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.