Poisson Probability Calculator Formula
Understand the math behind the poisson probability calculator. Each variable explained with a worked example.
Formulas Used
P(X = k)
probability = (pow(lambda, k) * pow(e, -lambda)) / factorial(k)Expected Value
expected = lambdaStandard Deviation
std_dev = sqrt(lambda)Variables
| Variable | Description | Default |
|---|---|---|
lambda | Average Rate (lambda) | 4 |
k | Number of Events (k) | 2 |
How It Works
How to Calculate Poisson Probability
Formula
P(X = k) = (lambda^k * e^(-lambda)) / k!
The Poisson distribution models the number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate lambda. Both the mean and variance equal lambda.
Worked Example
A call center receives an average of 4 calls per hour. What is the probability of exactly 2 calls in an hour?
- 01lambda = 4, k = 2
- 02P(X=2) = (4^2 * e^(-4)) / 2!
- 03= (16 * 0.01832) / 2
- 04= 0.29305 / 2
- 05= 0.14653
Frequently Asked Questions
When should I use the Poisson distribution?
Use it when counting events in a fixed interval (time, area, volume), events are independent, the rate is constant, and two events cannot occur at exactly the same instant. Examples: emails per hour, defects per meter, accidents per year.
How does Poisson relate to the binomial?
The Poisson distribution is the limit of the binomial as n approaches infinity and p approaches zero while np = lambda stays constant. It approximates the binomial well when n >= 20 and p <= 0.05.
Can lambda be a decimal?
Yes. Lambda represents the average rate and can be any positive real number. For example, 2.5 emails per minute is perfectly valid.
Ready to run the numbers?
Open Poisson Probability Calculator