What is Mathematical Expectation?

Mathematical expectation, or expected value, is the projected average outcome of a random event if you repeat it many times over a long period.

Consider a fair coin flip game where you bet $1 on heads. If it lands on heads, you win$1. If it lands on tails, you lose $1.

The expected value is calculated by multiplying each outcome by its probability and adding the products together:

$E(X) = (P(\text{heads}) \times \text{Gain}) + (P(\text{tails}) \times \text{Loss})$

$E(X) = \left(\frac{1}{2} \times \$1\right) + \left(\frac{1}{2} \times (-\$1)\right) = \$0.50 - \$0.50 = \$0.00$

An expected value of $0 means the game is perfectly fair. If the expected value is negative, you can expect to lose money in the long run.