The document provides an overview of geometric and Poisson distributions. It discusses how geometric distributions can model repeated trials until success, like passing a drivers test. It gives an example of calculating the probability of passing a drivers test on the 4th try. It then explains that Poisson distributions can model counting events that occur randomly in an interval, like accidents at an intersection. It provides an example of calculating the probability of 4 accidents occurring in a given month. Finally, it assigns homework problems related to these distributions.

1. Geometric & Poisson Distributions Chapter 4.3

2. Objectives Learn to find probabilities using the geometric distribution Learn to find probabilities using the Poisson distribution

3. Geometric Distributions Many actions in life are repeated until a success occurs: MCAS Drivers License test Safety tests in shop Can you think of others? If you graphed the number of people who passed, what do you think it would look like?

4. Geometric Distributions A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions: A trial is repeated until a success occurs. The repeated trials are independent of each other. The probability of success p is constant for each trial. The probability that the first success will occur on trial number x is P(x) = p(q) x-1 , where q = 1 - p

5. Drivers License Example In California, 23% of people who take the drivers license written exam pass. (LA Times Dec. 4, 2000) Find the probability that a person will pass on the 4 th try: P(4) = .23*.773 = .105003 = 10.5%

6. Poisson Distribution Sometimes you want to find the probability that a specific number of occurrences takes place within a given unit of time or space. What is the probability that a major hurricane will hit the mainland next year? What is the probability that it will snow more than 100 inches next winter?

7. Poisson Distribution The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions. The experiment consists of counting the number of times, x, an event occurs in a given interval. The interval can be an interval of time, area, or volume. The probability of the event occurring is the same for each interval. The number of occurrences in one interval is independent of the number of occurrences in other intervals.

8. Poisson Distribution The probability of exactly x occurrences in an interval is P(x) = μ x e - μ x! Where e is an irrational number approximately equal to 2.71828 and μ is the mean number of occurrences per interval unit.

9. Poisson Example The mean number of accidents per month at a certain intersection is 3. What is the probability that in any given month 4 accidents will occur at this intersection? P(4) = 3 4 (2.71828) -3 4! = .168

10. Poisson Table Look at Table 3 in Appendix B. A population count shows that there is an average of 3.6 rabbits per acre living in a field. Use a table to find the probability that 2 rabbits are found on any given acre of the field. X = 2, μ = 3.6 Answer is .1771

11. Homework P. 202 Do 1-9 together P. 194 10-22 evens