1. The Poisson Probability Distribution
Skill 27

2. Objectives
Use the Poisson distribution to compute the probability of the occurrence of events spread out over time or space.
Use the Poisson distribution to approximate the binomial distribution when the number of trials is large and the probability of success is small.

3. Poisson Probability Distribution
If we examine the binomial distribution as the number of trials n gets larger and larger while the probability of success p gets smaller and smaller, we obtain the Poisson distribution.
The Poisson distribution applies to accident rates, arrival times, defect rates, the occurrence of bacteria in the air, and many other areas of everyday life.

4. Poisson Probability Distribution
As with the binomial distribution, we assume only two outcomes: A particular event occurs (success) or does not occur (failure) during the specified time period or space.
We are interested in computing the probability of r occurrences in the given time period, space, volume, or specified interval.

5. r = number of successes
λ = mean number of successes over an interval

6. Example–Poisson Distribution
Pyramid Lake is located in Nevada on the Paiute Indian Reservation. 8-to-10-pound trout are not uncommon, and 12-to-15-pound trophies are taken each season.
The following information was given about the November catch for boat fishermen.
Total fish per hour = 0.667
Suppose you decide to fish Pyramid Lake for 7 hours during the month of November.

7. Example–Poisson Distribution
Use the information provided to find a probability distribution for r, the number of fish (of all sizes) you catch in a period of 7 hours.
Solution:
For fish of all sizes,
Mean success rate per hour is
 = 0.667/1 hour

8. Example–Solution Since we want to study a 7-hour interval
Rounded value;  = 4.7 for a 7-hour period.

9. Example–Solution
Since r is the number of successes (fish caught) in the corresponding 7-hour period and  = 4.7 for this period, use the Poisson distribution to get,

10. Example–Poisson Distribution
What is the probability that in 7 hours you will get 0, 1, 2, or 3 fish of any size?
Solution:

11. Example–Poisson Distribution
What is the probability that you will get four or more fish in the 7-hour fishing period?
Solution:
P (r  4) = P(4) + P(5) + … = 1 – P(0) – P(1) – P(2) – P(3)
 1 – 0.01 – 0.04 – 0.10 – 0.16
= 0.69
There is about a 69% chance that you will catch four or more fish in a 7-hour period.

12. Example–Poisson Distribution
The average number of homes sold by the Acme Realty Company is 2 homes per day. What is the probability that exactly 3 homes will be sold tomorrow?
Solution:
r = 3
λ = 2
There is about a 18.04% chance that the Acme Realty Company will sell exactly three homes tomorrow.

13. Example–Poisson Distribution
Suppose the average number of lions seen on a 1-day safari is five. What is the probability that tourists will see fewer than four lions on the next 1-day safari?
Solution:
λ = 5
r < 4
P(0) = .0067
P(1) = .0337
P(2) = .0842
P(3) = .1404
There is about a 26.5% chance that a tourist will see fewer than four lions on the next 1-day safari.

14. Example–Poisson Distribution
Suppose the average number of lions seen on a 1-day safari is five. What is the probability that tourists will see more than four lions on the next 1-day safari?
Solution:
λ = 5 and r > 4
P(r > 4) = 1 – [P(0) + P(1) + P(2) + P(3) + P(4)]
P(0) = .0067; P(1) = .0337; P(2) = .0842; and P(3) = .1404
P(4) = .1755
P(r > 4) = 1 – [.4405]
P(r > 4) = .5595
There is about a 55.95% chance that a tourist will see fewer than four lions on the next 1-day safari.

15. Example–Poisson Distribution
On a particular river, overflow floods occur once every 100 years on average. You are a historian studying the river, calculate the probability of the river overflowing more than three times in a 200-year period.
Solution:
λ = 1/100yrs
Need to find average in 200 years.
λ = 2/200yrs
r > 3 times
P(r > 3) = 1 – [P(0) + P(1) + P(2) + P(3)]

16. Example–Poisson Distribution
P(r > 3) = 1 – [P(0) + P(1) + P(2) + P(3)]
P(0) = .1353
P(1) = .2707
P(2) = .2707
P(3) = .1804
P(r > 3) = 1 – [.8571]
P(r > 3) = .1429
There is about a 14.29% chance that the river will overflow more than three times in a 200-year period.

17. 27: Poisson Probability Distribution
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