A fair six sided die is rolled once. You are paid the number shown in dollars, except that a 6 pays $20. What is the fair price of the game, as an exact fraction?

A fair six sided die is rolled once. You are paid the number shown in dollars, except that a 6 pays $20. What is the fair price of the game, as an exact fraction?

Approach: Price the plain version of the game first, then add the probability weighted size of the single deviation, rather than summing all six branches.

35/6. The five ordinary faces contribute 1 + 2 + 3 + 4 + 5 = 15 and the six pays 20, so the total across all faces is 35 and the expected value is 35/6, which is $5.833. The faster route prices a base case and adds the deviation: a plain die is worth 3.5, the six pays 14 more than its face, and that happens with probability 1/6, so the value is 3.5 plus 2.333, which is 3.5 + 14/6 = 5.833. Linearity of expectation is what allows the split, since the payoff is the face value plus a 14 dollar bonus on one face. It scales to any bet with many outcomes and a handful of exceptions, where summing every branch is slow and error prone.

Follow-up: The same game now pays $20 on a 6 and nothing on a 1. What is the fair price, and how much is the downside exception worth?

Key concepts: expected value, linearity of expectation, base case plus deviation.