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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.

This example illustrates how manufacturers can use probability distributions to estimate warranty costs. With an expected value of less than 1 failure per vehicle over 10 years, the manufacturer can better plan for replacement parts inventory and service center staffing.

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.

Sports analysts often use this calculation to project team performance. In this case, the team is expected to win approximately 3 games in the tournament, which helps coaches develop strategies and management set appropriate expectations for fans and sponsors.

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.

For sales managers, this expected value calculation helps with quota setting and commission planning. An average of 22.7 sales per month provides a baseline against which individual performance can be measured and appropriate incentives structured.

Example 4: Mean Number of Customer Service Calls

The following probability distribution shows the number of customer service calls a small business receives per hour during business hours:

Question: What is the mean number of customer service calls per hour?

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

μ = 0*0.05  +  1*0.15  +  2*0.25  +  3*0.30  +  4*0.20  +  5*0.05  =  2.6 calls per hour.

This information is useful for staffing a customer service team. With an average of 2.6 calls per hour, a manager can determine how many representatives are needed to maintain reasonable wait times and good service quality.

Example 5: Mean Daily Rainfall

The following probability distribution shows the amount of daily rainfall (in inches) during spring months in a particular region:

Question: What is the mean daily rainfall in this region during spring?

Solution: The mean daily rainfall is calculated as:

μ = 0*0.40  +  0.1*0.20  +  0.25*0.15  +  0.5*0.10  +  1.0*0.10  +  2.0*0.05  =  0.4875 inches.

This calculation helps meteorologists and city planners understand average precipitation patterns. With an expected value of 0.4875 inches of daily rainfall during spring, authorities can plan for appropriate drainage systems and farmers can determine optimal planting schedules.

Wrapping Up

Finding the mean of a probability distribution is a straightforward process that yields valuable information. To calculate the mean of any probability distribution:

1. Identify all possible values of the random variable
2. Multiply each value by its corresponding probability
3. Sum all the products to find the expected value

This statistical measure helps in forecasting, planning, and decision-making across many fields including finance, sports analytics, quality control, and business management. The mean of a probability distribution gives us a single representative value that summarizes what we can expect in the long run.

Bonus: Probability Distribution calculator

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