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(6) $ Probability of exactly 6 occurrences: 0.13226955027012

### Solution:

**$P(6)$ Probability of exactly 6 occurrences**

If using a calculator, you can enter $ \lambda = 4.6 $ and $ x = 6 $ 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 = 6 $ and $ \lambda = 4.6 $, we have
$$ P(6) = \frac{{e^{-4.6}} \cdot {4.6^6}}{6!} $$
Evaluating the expression, we have
$$ P(6) = 0.13226955027012 $$