# 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
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 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 than 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)