# Poisson Probability Distribution Calculator

An online calculator to calculate the Poisson probability distribution and the probabilities of "at least" and "at most" related to the binomials.

## Poisson Probability Distribution

The Poisson probability distribution calculator computes the probability that an event $A$ occurs $x$ times over a period of time (or space) knowing that this event occurs at an average event happens an average $\lambda$ times over that period.
$P(X = x) = \dfrac{e^{-\lambda}\lambda^x}{x!}$
The calculator below calculates the Poisson probability distribution $P(X = x)$ for any value of $x \ge 0$, given the average $\lambda$. The calculator below helps in investigating these distributions in various situations.
The same calculator also calculates the probability of "at least" $x$ given by $P(X \ge x)$ and "at most" $x$ given by $P(X \le x)$

Example 1
An event A occurs at an average of 4 times over a period of 24 hours.
a) What is the probability that event A occurs 5 times over a period of 24 hours?
b) What is the probability that event A occurs at most 5 times over a period of 24 hours?
c) What is the probability that event A occurs at least 5 times over a period of 24 hours?

Solution to Example 1
The average $\lambda = 4$ is over a period of of 24 hours. The probability to be calculated is over the same period. Hence
a)
$P(X = 5) = \dfrac{e^{-4}4^5}{5!} = 0.15629$
b) At most 5 times means $x$ is either $0, 1, 2 , 3, 4 \; \text{or} \; 5$ or $x \le 5$
$P(\text{at most 5 times}) = P( X = 0 \; or \; X = 1 \; or \; X = 2 \;$
$or \; X = 3 or \; X = 4 \; or \; X = 5 )$
Using the binomial formula, the probability may be written as
$P(X \le 5) = P(X = 0) + P(X = 1) + P(X = 2)$
$+ P(X = 3) + P(X = 4) + P(X = 5)$
$= 0.018315 + 0.073262 + 0.146525 + 0.195366 + 0.195366 + 0.156293$
$= 0.78513$
c)
At least 5 times means $x$ is equal or greater that 5.
$P(\text{at least 5 times}) = 1 - P(\text{at most 4 times})$
$= 1 - ( P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) )$
$= 1 - (0.018315 + 0.073262 + 0.146525 + 0.195366 + 0.195366 )$
$= 0.37116$

## How to use the calculator

1 - Enter $n$ and $p$ and $x$ and press "calculate". $n$ and $x$ are positive integers and $p$ real satisfying the conditions:
$\lambda \gt 0$ , $x \ge 0$

 $\lambda$ = 4 $x$ = 5 $P(X = x)$ = $P(X \ge x)$ = (at least) $P(X \le x)$ = (at most)

Poisson Probabilities Examples and Questions
Binomial Probabilities Examples and Questions