Five Properties of Poisson Distributions

1. There are only two possible outcomes for events – they can occur or not occur.
2. Each event must be random and independent of the others.
3. The occurrences must be uniformly distributed (see more in the uniform distributions section).
4. Probabilities are calculated on the number of occurrences, [latex]x[/latex], over a specific time interval.
5. An average number of occurrences over a given time period is denoted as [latex]\lambda[/latex].

Two Ways of Calculating the Probability of ‘x‘ Successes

If we want to calculate the probability of exactly [latex]x[/latex] events occurring per time period, we can:

1. Use the formula: [latex]P(x) = \frac{{\lambda}^x e^{-\lambda}}{x!}[/latex]
2. Use Excel: = POISSON.DIST([latex]x[/latex], [latex]\lambda[/latex], FALSE)

three examples of poisson distributions

Three examples of events that follow a Poisson Distribution:

1. The number of bad checks received by a bank per day.
2. The number of motor vehicle accidents in a busy street corner per week.
3. The number of shoppers arriving at a supermarket check-out counter per minute.

Calculating the probability of x events using the Formula (ViDEO)

Let us better understand the Poisson formula by working through an example.

Example 30.1.1

Problem Setup: On a busy Friday evening, the average number of shoppers arriving at a supermarket check-out counter waiting to be served by a cashier is 6 per minute. The arrivals follow a Poisson distribution.

Question: In any given minute, what is the probability that exactly 5 shoppers will arrive?

Written solution: Let us look at the formula a little more carefully now.

• There are really only 2 variables in this formula.
• One is [latex]\lambda[/latex], the average; in this example, which is 6.
• The other is [latex]x[/latex], the number for which we want to find the probability.  In this example it is 5.
• The other letter, [latex]e[/latex], is a constant. It is the base of the natural log and equal 2.71828.
• This gives: \[P(x=5) =\frac{6^5 e^{-6}}{5!}=\frac{(7,776) \times (0.002478752)}{ 120}=0.1606\]

Video solution: Click here to download the written solutions to the video below:

Conclusion: There is a 16.06% chance that 5 shoppers will arrive in any given minute.

Calculating the probability of x events using Excel (ViDEO)

Let us now use Excel’s POISSON.DIST function to solve the above example.

Example 30.1.2

Problem Setup: Let us now use Excel to solve for the following problem:

• An average of 6 shoppers arrive per minute at the till.
• Solve for the probability of exactly 5 shoppers will arriving in any given minute.

Solution: Use = POISSON.DIST(5, 6, 0). Click here to download the Excel solutions shown below:

Conclusion: There is a 16.06% chance that exactly 5 shoppers will arrive in any given minute.

Note: When using Excel’s POISSON.DIST(), we set cumulative = FALSE when calculating the probability of exactly [latex]x[/latex] events occurring. We will examine when we use cumulative = TRUE in later sections in this Chapter.

Calculating the probability of No events w/ Formula (Exercise)

In the first case, it was ‘a busy Friday evening.’ Let’s find out why in the next example.

Example 30.2.1

Problem Setup: An average of 6 shoppers arrive per minute at the check-out counter. 

Question: In any given minute, what is the probability that no shoppers arrive at the counter?

You Try: Solve the problem by placing the values in the correct positions below:

Conclusion: There is 0.25% chance that no shoppers will arrive in any given minute. In other words, there is almost NO chance that there are no shoppers arriving. This means that shoppers are almost constantly arriving, keeping the cashier busy.

Calculating the probability of No events using Excel (Exercise)

Let us now use Excel’s POISSON.DIST function to solve the above example. This time, I’m going to put you to work in setting up the Excel formula.

Example 30.2.2

Problem Setup: An average of 6 shoppers arrive per minute at a till, ready to check out. 

Question: In any given minute, what is the probability that there are no shoppers arriving at this check-out counter? Can you set up the problem in Excel?

You Try: Complete the exercise below to calculate the probability of no shoppers arriving:

Solution: Download the Excel solutions or click below to reveal the solutions:

Conclusion: There is 0.25% chance that no shoppers will arrive in any given minute.

Key Takeaways (EXERCISE)

Your Own Notes (EXERCISE)

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