# Expected Value of a Random Variable

## Expected Value of a Random Variable

Sum of the products of the values of a random variable by their probability.

In the case of a game of chance, a game is fair when the expected value is zero.

### 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 $$\frac{1}{6}$$; • If you win, your gain will be $$10 \times 0.50\space \times \frac{1}{6}$$, 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.

A more detailed way to calculate the expected value, E$$_{\textrm{M}}$$, in this specific case consists of considering all of the possible outcomes as well as the gains associated with them:
• E$$_{\textrm{M}}$$ = $$\left( 0 × \frac{5}{6} \right) + \left( 10 × 0.50 × \frac{1}{6} \right ) − 0.50 ≈ 0.33$$.