# 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:
\( \lambda \gt 0 \) , \( x \ge 0 \)

### More References and links

Poisson Probabilities Examples and QuestionsBinomial Probabilities Examples and Questions

addition rule of probabilities

multiplication rule of probabilities

probability questions

classical formula for probability

mutually exclusive events

Introduction to Probabilities

sample space

event

elementary statistics and probabilities .