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.

### Poisson Probability Calculator

### Answer:

$ P(5) $ Probability of exactly 5 occurrences: 0.15629345185053

### Solution:

**$P(5)$ 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
$$ P(5) = 0.15629345185053 $$