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.


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.

This game is therefore to your advantage.

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

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