# How to Find the Mean of a Probability Distribution (With Examples)

A probability distribution tells us the probability that a random variable takes on certain values.

For example, the following probability distribution tells us the probability that a certain soccer team scores a certain number of goals in a given game:

Note: The probabilities in a valid probability distribution will always add up to 1. We can confirm that this probability distribution is valid: 0.18 + 0.34 + 0.35 + 0.11 + 0.02 = 1.

To find the mean (sometimes called the “expected value”) of any probability distribution, we can use the following formula:

```Mean (Or "Expected Value") of a Probability Distribution:

μ = Σx * P(x)

where:
•x: Data value
•P(x): Probability of value```

For example, consider our probability distribution for the soccer team:

The mean number of goals for the soccer team would be calculated as:

μ = 0*0.18  +  1*0.34  +  2*0.35  +  3*0.11  +  4*0.02  =  1.45 goals.

The following examples show how to calculate the mean of a probability distribution in a few other scenarios.

### Example 1: Mean Number of Vehicle Failures

The following probability distribution tells us the probability that a given vehicle experiences a certain number of battery failures during a 10-year span:

Question: What is the mean number of expected failures for this vehicle?

Solution: The mean number of expected failures is calculated as:

μ = 0*0.24  +  1*0.57  +  2*0.16  +  3*0.03 =  0.98 failures.

### Example 2: Mean Number of Wins

The following probability distribution tells us the probability that a given basketball team wins a certain number of games in a tournament:

Question: What is the mean number of expected wins for this team?

Solution: The mean number of expected wins is calculated as:

μ = 0*.06  +  1*.15  +  2*0.17  +  3*0.24  +  4*.23  +  5*.09  +  6*.06  =  2.94 wins.

### Example 3: Mean Number of Sales

The following probability distribution tells us the probability that a given salesman will make a certain number of sales in the upcoming month:

Question: What is the mean number of expected sales for this salesman in the upcoming month?

Solution: The mean number of expected sales is calculated as:

μ = 10*.24  +  20*.31  +  30*0.39  +  40*0.06  =  22.7 sales.

### Bonus: Probability Distribution calculator

You can use this calculator to automatically calculate the mean of any probability distribution.