**Poisson Experiment**

A **poisson experiment **is an experiment that has the following properties:

- There are two possible outcomes of the experiment: success or failure
- The average number of successes (μ) that occurs in a specific interval is known
- The probability that a success will occur increases as the interval increases

**The Poisson Distribution**

A **Poisson random variable** is the number of successes, denoted x, that results from a Poisson experiment, and it follows a **Poisson distribution**.

A Poisson distribution has the following properties:

- The mean of the distribution is
**μ**=*n***p* - The variance (
**σ**) of the distribution is also^{2}**μ** - The standard deviation of the distribution is
**σ**= √σ^{2}

To find the probability of x successes in a Poisson experiment, given that we know the average number of successes (μ) in a particular interval, we can use the following formula:

P(x successes) = (e^{-μ}) (μ^{x}) / x!

where *e* is equal to about 2.71828.

Let’s walk through some examples to gain a better understanding of the Poisson distribution.

**Examples Using the Poisson Distribution**

**Example 1: A hardware store sells 3 hammers per day on average. What is the probability that they will sell 5 hammers on a given day?**

**Step 1: Identify the mean number of hammers sold per day ( μ) and the number of hammers we’re interested in analyzing (x).**

*μ *= 3

*x *= 5

**Step 2: Plug these numbers into the Poisson formula or a Poisson calculator**

**Using the formula:**

P(sell 5 hammers) = (e^{-μ}) (μ^{x}) / x!

P(sell 5 hammers) = (e^{-3}) (3^{5}) / 5!

P(sell 5 hammers) = (e^{-3}) (3^{5}) / 5! = (.049787) * (243) / (120) =** .10082**

**Using the calculator:**

Plug the following numbers into the Poisson Distribution Calculator:

The probability that the hardware store sells 5 hammers on a given day is equal to **0.10082**.

**Example 2: A certain grocery store sells 15 cans of tuna per day on average. What is the probability that this store sells more than 20 cans of tuna in a given day?**

**Step 1: Identify the mean number of cans sold per day ( μ) and the number of cans we’re interested in analyzing (x).**

*μ *= 15

*x *= 20

**Step 2: Since we are finding a cumulative probability (i.e. the combined probability that we sell 21 cans or 22 cans or 23 cans…), it’s easier to use the Poisson distribution calculator.**

Plug the following numbers into the Poisson Distribution Calculator:

The probability that this store sells *more *than 20 cans of tuna in a given day is **0.08297**.

**Example 3: A certain sporting goods store sells seven basketballs per day on average. What is the probability that this store sells four or less basketballs in a given day?**

**Step 1: Identify the mean number of basketballs sold per day ( μ) and the number of basketballs we’re interested in analyzing (x).**

*μ *= 7

*x *= 4

**Step 2: Since we are finding a cumulative probability (i.e. the combined probability that we sell 4 or less basketballs), it’s easier to use the Poisson distribution calculator.**

Plug the following numbers into the Poisson Distribution Calculator:

The probability that this store sells four or less basketballs in a given day is **0.17299**.