You learned in earlier chapters that you can convert a percentage to a decimal and multiply by a number to find that percentage of the number. You’ll use that same skill to find the expected value of a variable, $X$.

The expected value is a way to describe what you should expect from a situation with probabilities. It’s a weighted result: outcomes with higher probability count more than outcomes with lower probability.

Now, let’s make one important point clear:

The expected value is not the same as the mean value.

• The mean (average) describes the average of a set of numbers.
• The expected value describes the value you expect when outcomes happen with certain probabilities.

You’ll use expected value when the problem gives probabilities and asks for what you can expect overall.

Now, let’s see how to calculate it.

Calculating the expected value

To find the expected value, calculate the sum of each possible value multiplied by its probability.

Let’s use an example.

Day of the week	Chance of rain	Predicted height of rainfall (inches)
Monday	20%	2
Tuesday	10%	1
Wednesday	0%	0.5
Thursday	35%	0.5
Friday	35%	1

Using the table of data above and assuming that it will rain one day next week, what is the expected height of rainfall in inches?

Go row by row:

• Convert the chance of rain to a decimal (this is the probability).
• Multiply that probability by the predicted rainfall amount (this is the possible outcome).
• Add all the products.

You can think of it like this: “There’s a 20% chance of 2 inches on Monday, a 10% chance of 1 inch on Tuesday, …”

$$(0.2\ast 2)+(0.1\ast 1)+(0\ast 0.5)+(0.35\ast 0.5)+(0.35\ast 1)=1.025$$

The expected height of rainfall on any given day is $1.025$ inches.

The expected value formula

Here is the general formula. $E(X)$ is the expected value, each $x$ is a possible value, and each $P$ is the probability of that value.

$$E(X)=x_1\ast P_1+x_2\ast P_2+x_3\ast P_3+\dots$$

In the rainfall example, $x_1$ was $2$ inches and $P_1$ was $20\%$ (or $0.20$). You multiplied each outcome by its probability and added the results, exactly as the formula shows.

Now try a simpler example.

What is the expected value of $X$ given it has the following values and probabilities?

4	22%
2	45%
1	10%
6	7%
7.5	14%
10	2%

A quick check for your answer

In these problems, the probabilities should add up to $100\%$. When they do, the expected value should fall between the smallest and largest values in the table.

• If your expected value is smaller than the smallest value or larger than the largest value, check your work.
• A common mistake is forgetting to convert a percent to a decimal.