A fair die pays you its face value in dollars. What is the fair price to play, and what is the variance of the payout?

How to answer

Fair price = $3.50 = (1+2+3+4+5+6)/6. Variance = E[X²] − E[X]² = (1+4+9+16+25+36)/6 − 3.5² = 91/6 − 12.25 = 35/12 ≈ 2.917. The fair price is just E[X]; the variance quantifies the risk you'd charge a spread to bear.

Key idea: Price = E[X]; variance = E[X²] − (E[X])².