Solution
Fair Price of a Squared Roll
Show the problem again
You pay an entry fee to play the following game: a fair six-sided die is rolled once, and you receive k² dollars, where k is the face shown. What entry fee makes the game fair, meaning your expected net gain is exactly zero? Give the exact value in dollars as a reduced fraction (a decimal rounded to two places is also accepted).
Worked solution
The answer is 91/6 dollars, about $15.17. A fair fee equals the expected payout E[k²] = (1 + 4 + 9 + 16 + 25 + 36)/6 = 91/6. Note that this is not (3.5)² = 12.25: the expected value of a square exceeds the square of the expected value by the variance, here 35/12. Paying the square of the average roll would hand the player an edge of 35/12 dollars per game.
Source: Standard expected-value exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.