The Poisson Probability Calculator can calculate the probability of an event occurring in a given time interval.  Before using the calculator, you must know the average number of times the event occurs in the time interval.  The symbol for this average is $\lambda$, the greek letter lambda.  You also need to know the desired number of times the event is to occur, symbolized by x. If you’d like to construct a complete probability distribution based on a value for $\lambda$ and x, then go ahead and take a look at the Poisson Distribution Calculator.  It will calculate all the poisson probabilities from 0 to x.

$P(5)$ Probability of exactly 5 occurrences: 0.15629345185053
$P(5)$ Probability of exactly 5 occurrences
If using a calculator, you can enter $\lambda = 4$ and $x = 5$ into a poisson probability distribution function (PDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 5$ and $\lambda = 4$, we have $$P(5) = \frac{{e^{-4}} \cdot {4^5}}{5!}$$ Evaluating the expression, we have $$P(5) = 0.15629345185053$$