1 Poisson Probability Models The Poisson experiment typically models situations where rare events occur over a fixed amount of time or within a specified region Examples ◦ The number of cellphone calls per minute arriving at a cellphone tower. ◦ The number of customers per hour using an ATM ◦ The number of concussions per game experienced by the participants.

3 ◦ Properties of the Poisson experiment 1)The number of successes (events) that occur in a certain time interval is independent of the number of successes that occur in another time interval. 2)The probability of a success in a certain time interval is  the same for all time intervals of the same size,  proportional to the length of the interval. 3)The probability that two or more successes will occur in an interval approaches zero as the interval becomes smaller. Poisson Experiment

4 The Poisson Random Variable ◦ The Poisson random variable X is the number of successes that occur during a given time interval or in a specific region Probability Distribution of the Poisson Random Variable.

Poisson Prob Dist =1

Poisson Prob Dist =5

7 Example 1 ◦ Cars arrive at a tollbooth at a rate of 360 cars per hour. ◦ What is the probability that only two cars will arrive during a specified one-minute period?  The probability distribution of arriving cars for any one- minute period is Poisson with = 360/60 = 6 cars per minute. Let X denote the number of arrivals during a one-minute period.

8 ◦ Example 1 (cont.) ◦ What is the probability that at least four cars will arrive during a one-minute period? ◦ P(X>=4) = 1 - P(X<=3) = =.849

Example 2 (Yecchh!) The Food and Drug Administration sets a Food Defect Action Level (FDAL) for the maximum amounts of various foreign substances that can be in the food we eat. The FDAL for insect fragments in peanut butter is 0.3 insect fragments (larvae, eggs, body parts, etc.) per gram.FDAL Suppose the brand of peanut butter used in your dorm’s cafeteria has 0.3 insect fragments per gram.

Example 2 (cont.) In a 5-gram helping of your dorm’s peanut butter - What is the probability of 2 insect fragments in the peanut butter? What is the probability of 3 or more insect fragments in the peanut butter?