Poisson Probability CalculatorFree online math calculator — instant results, no sign-up
Calculate a Poisson probability for a specified event count.
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About the Poisson Probability Calculator
The Poisson Probability Calculator finds the probability of observing exactly a specified number of events when the expected count for the same interval is known. With an expected event count of 4 and an observed event count of 3, it returns 0.1953668148, or approximately 19.54%. This is an exact-count probability, not the probability of observing at most or at least that many events.
- Expected event count
- The mean number of events expected in the chosen time, area, or other observation interval, measured in events per interval and defaulting to 4.
- Observed event count
- The exact nonnegative integer count whose probability you want to calculate, measured in events within that same interval and defaulting to 3.
Formula and method
P(X = k) = e^(−λ) × λ^k / k!; with λ = 4 and k = 3, P(X = 3) = e^(−4) × 4^3 / 3! = e^(−4) × 64 / 6 ≈ 0.1953668148.
- XThe random number of events occurring in the observation interval.
- kThe specified observed event count, a nonnegative integer; the default is 3.
- λThe expected event count for the same interval, a nonnegative mean; the default is 4.
- eEuler's number, the base of the natural exponential function.
- k!The factorial of k, with 0! defined as 1.
- P(X = k)The probability of exactly k events occurring in the interval.
Evaluate the Poisson probability mass function at the specified count: multiply e^(−λ) by λ^k, then divide by k!. For the default inputs, 4^3 = 64 and 3! = 6, so e^(−4) × 64 / 6 gives approximately 0.1953668148. For large inputs, a numerically stable implementation can evaluate the equivalent log-probability before exponentiating rather than computing large powers and factorials directly.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Expected event count4
- Observed event count3
Results
- Result0.1953668148
How to use the Poisson Probability Calculator
- 1Choose the observation interval so that the expected and observed counts refer to the same time span, area, or exposure.
- 2Enter the Expected event count; leave it at 4 to reproduce the default example.
- 3Enter the Observed event count as a nonnegative integer; leave it at 3 for the default example.
- 4Run the calculation to evaluate the probability of exactly the entered observed count.
- 5Read the Result: the default value of 0.1953668148 corresponds to approximately a 19.54% chance of exactly 3 events.
Reading your result
The Result row reports P(X = k), a unitless probability between 0 and 1. With an expected event count of 4 and an observed event count of 3, Result = 0.1953668148 means that exactly 3 events would occur in approximately 19.54% of comparable intervals under the Poisson model. It is not a cumulative probability or a measure of how likely the assumed mean is to be correct.
Limitations
The Poisson model assumes independent events and a fixed mean for the defined observation interval; a homogeneous Poisson process additionally assumes a constant event rate. Clustering, dependence, or changing exposure can make this model unsuitable. A Poisson distribution has equal mean and variance, so substantial departures from that relationship may indicate a poor fit. The expected count must be finite and nonnegative, while the observed count must be a nonnegative integer. Very small probabilities may round to zero, and uncertainty in an estimated expected count is not included.
Common questions
Does the default result give the probability of 3 or fewer events?+
No. With an expected count of 4, the result 0.1953668148 is only the probability of exactly 3 events. The probability of 3 or fewer requires summing the Poisson probabilities for all counts from zero through 3.
Can the expected event count be a decimal even though the observed count must be an integer?+
Yes. The expected count is an average across comparable intervals and can be fractional. The observed count represents a number of discrete events, so it must be a nonnegative integer.
Can I enter an event rate directly as the expected event count?+
Only if that rate already corresponds to the chosen observation interval. Otherwise, multiply the rate by the interval's duration or exposure, using compatible units, to obtain the expected count before entering it.
What happens when the expected event count is zero?+
A Poisson model with a zero expected count assigns probability 1 to zero observed events and probability 0 to any positive observed count. This boundary case should be handled directly rather than relying on an ambiguous evaluation of 0^0.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.