Poisson Distribution Calculator
The Poisson Calculator makes it easy to compute individual and cumulative Poisson probabilities. For help in using the calculator, read the Frequently-Asked Questions or review the Sample Problems.
To learn more about the Poisson distribution, read Stat Trek's lesson on the Poisson distribution.
- Select a probability from the dropdown box.
- Enter values for μ and x (in the first two textboxes).
- Click Calculate to compute probabilities.
Frequently-Asked Questions
Instructions: To find the answer to a frequently-asked question, simply click on the question.
What is a Poisson experiment?
A Poisson experiment examines the number of times an event occurs during a specified interval. The interval could be anything - a unit of time, length, volume, etc. We might, for example, ask how many customers visit a store each day, or how many home runs are hit in a season of baseball.
A Poisson experiment has the following characteristics:
- The average rate of success is known.
- The probability that a single success will occur during a short interval is proportional to the size of the interval.
- The probability that a success will occur within a short interval is independent of successes that occur outside the interval.
- The probability of more than one success occurring within a very short interval is small.
What is a Poisson distribution?
The number of successes in a Poisson experiment is referred to as a Poisson random variable. A Poisson distribution is a probability distribution of a Poisson random variable.
For example, suppose we know that a receptionist receives an average of 1 phone call per hour. We might ask: What is the likelihood that she will get 0, 1, 2, 3, or 4 calls next hour. If we treat the number of phone calls as a Poisson random variable, the various probabilities can be calculated, as shown in the table below.
| Number of Phone Calls | Probability | Cumulative probability |
|---|---|---|
| 0 | 0.368 | 0.368 |
| 1 | 0.368 | 0.736 |
| 2 | 0.184 | 0.919 |
| 3 | 0.061 | 0.981 |
| 4 | 0.015 | 0.996 |
| 5 | 0.003 | 0.999 |
| 6 or more | 0.001 | 1.00 |
Taken together, the values for the Poisson random variable and their associated probabilities represent a Poisson distribution.
What is a Poisson random variable?
A Poisson random variable refers to the number of successes in a Poisson experiment.
We might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood next hour that she will receive 4 phone calls next hour.
If we treated this as a Poisson experiment, then the value of the Poisson random variable would be 4. (And the average rate of success would be 1.)
What is the average rate of success?
The average rate of success refers to the average number of successes that occur over a particular interval in a Poisson experiment. It is the mean of the Poisson distribution. In many texts, the mean (μ) of a Poisson distribution is referred to as lambda (λ).
We might be interested in the number of phone calls received in an hour by a receptionist. Suppose she received 1 phone call per hour on average.
If we treated this as a Poisson experiment, each phone call would represent a success in the experiment. The average rate of success over a 1-hour period would be 1 phone call. Note, however, that our experiment might involve a different unit of time. Suppose we focused on the number of calls during a 30-minute time period. Then, the average rate of success would be 0.5 calls per half hour. Similarly, if we focused on a 2-hour time period, the average rate of success would be 2 calls per 2 hours.
What is a Poisson probability?
A Poisson probability refers to the probability of getting EXACTLY n successes in a Poisson experiment. Here, n would be a Poisson random variable.
For instance, we might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood next hour that she will receive EXACTLY 3 phone calls? The probability of getting EXACTLY 3 phone calls in the next hour would be an example of a Poisson probability, denoted by P(X=3). (Note: In this example, the Poisson probability is equal to 0.061.)
What is a cumulative Poisson probability?
A cumulative Poisson probability refers to the probability that the Poisson random variable (X) falls within a certain range. For example, consider the probability of getting AT MOST n successes in a Poisson experiment. Here, n would be a Poisson random variable. And the cumulative Poisson probability would be the probability that n falls within the range of 0 and n.
For instance, we might be interested in the number of phone calls received in an hour by a receptionist. Suppose we knew that she received 1 phone call per hour on average. We might ask: What is the likelihood that she will receive AT MOST 1 phone call next hour? The probability of getting AT MOST 1 phone call in the next hour would be an example of a cumulative Poisson probability.
Note: The cumulative Poisson probability in this example is equal to the probability of getting zero phone calls PLUS the probability of getting one phone call. Thus, the cumulative Poisson probability would equal 0.368 + 0.368 or 0.736. (For details, see the question above: What is a Poisson distribution.)
Notation associated with cumulative Poisson probability is best explained through illustration. The probability of getting LESS THAN 1 phone call is indicated by P(X < 1); the probability of getting AT MOST 1 phone call is indicated by P(X ≤ 1); the probability of getting AT LEAST 1 phone call is indicated by P(X ≥ 1); the probability of getting MORE THAN 1 phone call is indicated by P(X > 1).
What is the standard deviation of a Poisson distribution?
The variance (σ2) of a Poisson distribution equals the mean (μ) of the Poisson distribution. The standard deviation (σ) is the square root of the variance; so, the standard deviation of a Poisson distribution equals the square root of the mean of the Poisson distribution.
σ = sqrt(μ)
The standard deviation measures the typical distance of a Poisson random variable from the mean.
Sample Problems
-
Historically, schools in a Dekalb County close 3 days each year, due to snow.
What is the probability that schools in Dekalb County will close for 4 days
next year?
Solution:
We are interested in the probability associated with a single Poisson random variable (x), so we select P(X=x) from the Probability dropdown box. And we know the following:- The Poisson random variable is 4.
- The average rate of success is 3. Here, we define a "success" as a school closing. Since the schools have closed historically 3 days each year due to snow, the average rate of success is 3. Therefore, we plug those numbers into the Poisson Calculator and hit the Calculate button.
The calculator reports that the Poisson probability is 0.16803. That is the probability of getting EXACTLY 4 school closings due to snow, next winter. Note that the calculator also reports other probabilities. For example, the calculator reports that the probability of getting AT MOST 4 school closings in the coming year is 0.815.
-
An expert typist makes, on average, 2 typing errors every 5 pages. What is the
probability that the typist will make at most 5 errors on the next fifteen
pages?
Solution:
We are interested in the probability associated with a single Poisson random variable (x), so we select P(X=x) from the Probability dropdown box. And we know the following:- The Poisson random variable is 5.
- The average rate of success 6. This may require a little explanation. We know that the average rate of success is 2 errors for every five pages. However, this problem calls for typing three times as many pages, so we would expect the typist to make three times as many errors, on average. Therefore, average rate of success is 3 x 2, which equals 6.
Therefore, we plug those numbers into the Poisson Calculator and hit the Calculate button.
The calculator reports that the P(X<5) is 0.44568. In other words, the probability that the typist makes no more than 5 errors is 0.44568. Note that the calculator also displays other probabilities. For example, the probability that the typist makes EXACTLY 5 errors is 0.16062.
-
Over his major league baseball career, Ted Williams averaged 1.158 hits per game. If you selected a random game
from his career, what is the probability that Ted Williams would get at least one hit but not more than two
hits in that game?
Solution:
We are interested in the probability associated with a range of Poisson random variables (x1 and x2), so we select P(x1 ≤ X ≤ x2) from the Probability dropdown box. And we know the following:- The minimum Poisson random variable (x1) is 1, and the maximum Poisson random variable (x2) is 2.
- The average rate of success 1.158, since Ted Williams averaged 1.158 hits per game over his career.
Therefore, we plug those numbers into the Poisson Calculator and hit the Calculate button.
The calculator reports that the P(1 ≤ X ≤ 2) is 0.31411. In other words, there is about a 31% chance that Ted Williams got at least one but not more than 2 hits in any randomly selected game.