Example

When rolling a fair die with six faces numbered 1 to 6, you bet $0.50 on the 6, hoping to win 10 times your bet. In this case, the expected value is calculated like this:

• the probability of rolling a 6 is [latex]\frac{1}{6}[/latex];
• If you win, your gain will be [latex]10 \times 0.50\space $ \times \frac{1}{6}[/latex], which is about $0.83;
• Because you spend $0.50 on each roll as you bet, your net gain will be $0.33, because: 0.83 – 0.50 = 0.33, on average.

This game is therefore to your advantage. A more detailed way to calculate the expected value, E[latex]_{\textrm{M}}[/latex], in this specific case consists of considering all of the possible outcomes as well as the gains associated with them:

• E[latex]_{\textrm{M}}[/latex] = [latex]\left( 0 × \frac{5}{6} \right) + \left( 10 × 0.50 × \frac{1}{6} \right ) − 0.50 ≈ 0.33[/latex].