Is 10-to-1 enough for picking four winners

Question

A bookmaker offers a parlay: you must correctly pick the winners of four separate games, each of which you regard as a 50-50 contest, and if all four picks are right you are paid at odds of 10 to 1. Should you take the bet?

Short answer

No. Winning all four has probability $(\tfrac12)^4=\tfrac1{16}$ , so fair odds would be 15 to 1. At 10 to 1 a $1 stake returns $11/16\approx\text{\char36}0.69$ on average, an expected loss of about 31 cents per dollar.

Adapted from Crack, Heard on the Street, Q4.19, p.41; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited
Sample Spaces, Events, and the Probability Axioms

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