Skill: Calculate expected value

The expected value is the probability-weighted mean of all possible outcomes. It tells you the average result you would expect over many repetitions.

Use the formula

$$E(X)=\sum xP(X=x)$$

This means:

1. Multiply each outcome $x$ by its probability.
2. Add all the products.
3. Check that the probabilities add to $1$.

Worked example

A game has the following possible winnings:

Winnings $x$	Probability $P(X=x)$
$0	$0.5$
$5	$0.3$
$20	$0.2$

First, check the probabilities:

$$0.5+0.3+0.2=1$$

Now multiply each outcome by its probability:

$$\begin{aligned} E(X) &=(0)(0.5)+(5)(0.3)+(20)(0.2)\\ &=0+1.5+4\\ &=5.5 \end{aligned}$$

Therefore, the expected value is

$$\boxed{\$5.50}$$

This does not mean you will win exactly 5.50$in one game. It means that if the game were played many times, the average winnings per game would approach$$5.50$.