A probability distribution tells us the probability that a 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:

To find the expected value of a probability distribution, we can use the following formula:

μ = Σx * P(x)

where:

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

For example, the expected 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 step-by-step example shows how to calculate the expected value of a probability distribution on a TI-84 calculator.

Step 1: Enter the Data

First, we will enter the data.

Press Stat, then press EDIT. Then enter the data values in column L1 and their probabilities in L2:

Step 2: Multiply the Two Columns

Next, we will multiply columns L1 and L2.

Highlight the top of column L3 and type in the following formula: L1 * L2

Use these steps to enter this formula:

• Press Stat, then press 1.
• Press the multiplication x button.
• Press Stat, then press 2.

Once you press Enter, the following values will appear in column L3:

Step 3: Find the Expected Value

Lastly, use the following steps to find the expected value of the probability distribution:

• Press 2nd and then press MODE to return to the home screen.
• Press 2nd and then press STAT. Scroll over to “MATH” and then press 5.
• Press 2nd and then press 3.
• Press the ) button.

Once you press Enter, the expected value will be displayed:

The expected value turns out to be 1.45.

Notice that this matches the expected value that we calculated by hand at the beginning of the article.