Poisson Distribution Calculator: Online Statistical Table
The Poisson Calculator makes it easy to compute individual and cumulative
Poisson probabilities. For help in using the calculator, read the
FrequentlyAsked Questions or review the
Sample Problems.
To learn more about the Poisson distribution, read Stat Trek's
tutorial on the Poisson distribution.
Instructions: To find the answer to a frequentlyasked
question, simply click on the question. If none of the questions addresses your
need, refer to Stat Trek's tutorial
on the Poisson distribution or visit the
Statistics Glossary. Online help is just a mouse click away.
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.
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, then the average rate
of success over a 1hour 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 30minute time period. Then, the average rate of
success would be 0.5 calls per half hour. Similarly, if we focused on a 2hour
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.
(Note: The Poisson probability in this example is equal to 0.061. See the above
table.)
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 (see the above table for
details).
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).

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 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 0.168. That is the probability of getting EXACTLY 4 school
closings due to snow, next winter. (The calculator also reports the cumulative
probability  the probability of getting AT MOST 4 school closings in the
coming year. The cumulative probability 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 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 < is 0.446. In other words, the probability
that the typist makes no more than 5 errors is 0.446. (Note that the calculator
also displays the Poisson probability  the probability that the typist makes
EXACTLY 5 errors. The Poisson probability is 0.161.)