### Introduction

There are two main characteristics of a Poisson experiment.

- The Poisson probability distribution gives the probability of a number of events occurring in a
*fixed interval*of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages. - The Poisson distribution may be used to approximate the binomial if the probability of success is
*small*(such as .01) and the number of trials is*large*(such as 1,000). You will verify the relationship in the homework exercises.*n*is the number of trials, and*p*is the probability of a*success*.

The random variable *X* = the number of occurrences in the interval of interest.

### Example 4.25

The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in five minutes. The time interval of interest is five minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in five minutes is three?

Let *X* = the number of loaves of bread put on the shelf in five minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, then the average number of loaves put on the shelf in five minutes is $\left(\frac{5}{30}\right)$(12) = 2 loaves of bread.

The probability question asks you to find *P*(*x* = 3).

The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes?

### Example 4.26

A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let *X* = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day. Write a mathematical statement for the probability question.

*P*(*x* < 5)

An electronics store expects to have 10 returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day. State the probability question mathematically.

### Example 4.27

You notice that a news reporter says "uh," on average, two times per broadcast. What is the probability that the news reporter says "uh" more than two times per broadcast?

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

a. What is the interval of interest?

a. one broadcast

b. What is the average number of times the news reporter says "uh" during one broadcast?

b. 2

c. Let *X* = ________. What values does *X* take on?

c. Let *X* = the number of times the news reporter says "uh" during one broadcast.

*x*= 0, 1, 2, 3, . . .

d. The probability question is *P*(______).

d. *P*(*x* > 2)

An emergency room at a particular hospital gets an average of five patients per hour. A doctor wants to know the probability that the ER gets more than five patients per hour. Give the reason why this would be a Poisson distribution.