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# What is a Probability Distribution Table? (Definition & Example)

A probability distribution table is a table that displays the probability that a random variable takes on certain values.

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

The left-hand column shows the number of goals and the right-hand column tells us the probability that the team will score this number of goals.

For example:

• The probability that the team scores exactly 0 goals is 0.18.
• The probability that the team scores exactly 1 goal is 0.34.
• The probability that the team scores exactly 2 goals is 0.35.

And so on.

### Properties of a Probability Distribution Table

A probability distribution table has the following properties:

1. All probabilities must add up to 1.Â

For a probability distribution table to be valid, all of the individual probabilities must add up to 1. We can verify that the previous probability distribution table is valid:

Sum of probabilities = 0.18 + 0.34 + 0.35 + 0.11 + 0.02 = 1.

2. The mean can be calculated.

The formula to calculate the mean of a given probability distribution table is:

Î¼ = Î£x * P(x)

where:

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

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

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

Î¼ = 0*0.18Â  +Â  1*0.34Â  +Â  2*0.35Â  +Â  3*0.11Â  +Â  4*0.02Â  =Â Â 1.45Â goals.

3. The standard deviation can be calculated.

The formula to calculate the standard deviation of a given probability distribution table is:

Ïƒ = âˆšÎ£(xi-Î¼)2Â * P(xi)

where:

• xi:Â The ithÂ value
• Î¼:Â The mean of the distribution
• P(xi):Â The probability of the ithÂ value

For example, hereâ€™s how to calculate the standard deviation of goals scored by the soccer team:

The standard deviation is the square root of the sum of the values in the third column:

Standard deviation = âˆš(.3785 + .0689 + .1059 + .2643 + .1301)Â =Â 0.9734

### How to Visualize a Probability Distribution Table

The easiest way to visualize the values in a probability distribution table is by using a histogram, which displays the values of the random variable along the x-axis and the probability of those values along the y-axis:

This lets us quickly visualize the probability values from the table.

In particular, we can see that there is a high probability that the team scores 2 goals or less while there is a tiny probability that the team scores as many as 4 goals.