Practice problems for the Poisson distribution

This post has practice problems on the Poisson distribution. For a good discussion of the Poisson distribution and the Poisson process, see this blog post in the companion blog.

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Practice Problems

Practice Problem 1
Two taxi arrive on average at a certain street corner for every 15 minutes. Suppose that the number of taxi arriving at this street corner follows a Poisson distribution. Three people are waiting at the street corner for taxi (assuming they do not know each other and each one will have his own taxi). Each person will be late for work if he does not catch a taxi within the next 15 minutes. What is the probability that all three people will make it to work on time?

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Practice Problem 2
A 5-county area in Kansas is hit on average by 3 tornadoes a year (assuming annual Poisson tornado count). What is the probability that the number of tornadoes will be more than the historical average next year in this area?

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Practice Problem 3
A certain airline estimated that 0.8% of its customers with purchased tickets fail to show up for their flights. For one particular flight, the plane has 500 seats and the flight has been fully booked. How many additional tickets can the airline sell so that there is at least a 90% chance that everyone who shows up will have a seat?

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Practice Problem 4
A life insurance insured 9000 men aged 45. The probability that a 45-year old man will die within one year is 0.0035. Within the next year, what is the probability that the insurance company will pay between 30 and 33 claims (both inclusive) among these 7000 men?

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Practice Problem 5
In a certain manuscript of 1000 pages, 300 typographical errors occur.

• What is the probability that a randomly selected page will be error free?
• What is the probability that 10 randomly selected pages will have at most 3 errors?

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Practice Problem 6
Trisomy 13, also called Patau syndrome, is a chromosomal condition associated with severe intellectual disability and physical abnormalities in many parts of the body. Trisomy 13 occurs , on the average, once in every 16,000 births. Suppose that in one country, 100,000 babies are born in a year. What is the probability that at most 3 births will develop this chromosomal condition?

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Practice Problem 7
Traffic accidents occur along a 50-mile stretch of highway at the rate of 0.85 during the hour from 5 PM to 6 PM. Suppose that the number of traffic accidents in this stretch of highway follows a Poisson distribution. The department of transportation plans to observe the traffic flow in this stretch of highway during this hour in a two-day period. What is the probability that more than three accidents occur in this observation period?

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Practice Problem 8
The odds of winning the Mega Million lottery is one in 176 million. Out of 176 million lottery tickets sold, what is the probability of having no winning ticket? What is the exact probability model in this problem?

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Answers

Practice Problem 1

• $1-5e^{-2}=0.323323584$

Practice Problem 2

• $1-13e^{-3}=0.352768111$

Practice Problem 3

• Can oversell by 2 tickets.

Practice Problem 4

• 0.278162459

Practice Problem 5

• 0.740818221
• 0.647231889

Practice Problem 6

• 0.130250355 (using Poisson)
• 0.130242377 (using Binomial)

Practice Problem 7

• 0.093189434

Practice Problem 8

• 0.367879441

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$\copyright \ 2015 \text{ by Dan Ma}$

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