## Practice Problem Set 3 – The Big 3 Discrete Distributions

This post presents exercises on the big 3 discrete distributions – binomial, Poisson and negative binomial, reinforcing the concepts discussed in several blog posts (here and here).

A previous problem set on Poisson and gamma is found here.

A previous problem set on Poisson distribution is found here.

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 Practice Problem 3-A The amount of damage from an auto collision accident is modeled by an exponential distribution with mean 5. Ten unrelated auto collision claims are examined by an insurance adjuster. What is the probability that five of the claims will have damages exceeding the mean damage amount? What is the probability that at most two of the claims will have damages exceeding the mean damage amount? What is the expected number of claims with damages exceeding the mean?

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 Practice Problem 3-B The jackpot of the Powerball lottery can sometimes be in the hundreds of millions dollars. The odds of winning the jackpot are one in 292 million. However, there are prizes other than the jackpot (some of the lesser prizes are $100 and$7). The odds of winning a prize in Powerball are one in 24.87. A Powerball player buys one ticket every month for a year. What is the probability of winning at least one prize? What is the probability of winning at least two prizes? What is the probability of winning at least three prizes? What is the probability of winning at least four prizes? See here for the calculation of Powerball winning odds.

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 Practice Problem 3-C According to a poll conducted by AAA, 94% of teen drivers acknowledge the dangers of texting and driving but 35% admitted to doing it anyway. In a random sample of 20 teen drivers, what is the probability that exactly five of the teen drivers do texting while driving? what is the probability that more than five of the teen drivers do texting while driving?

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 Practice Problem 3-D According to aviation statistics in the commercial airline industry, approximately one in 225 bags or luggage that are checked is lost. A business executive will be flying frequently next year and will be checking 100 bags or luggage during that one year. Determine the probability that the business executive will not lose any bags or luggage during his travel. Determine the probability that the business executive will lose one or two bags or luggage during his travel.

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 Practice Problem 3-E A large group of insured drivers are classified as high risk and low risk. About 10% of the drivers in this group are considered high risk while the remaining 90% are considered low risk drivers. The number of auto accidents in a year for a high risk driver in this group is modeled by a binomial distribution with mean 0.8 and variance 0.64. The number of auto accidents in a year for a low risk driver is modeled by a binomial distribution with mean 0.4 and variance 0.36. Suppose that an insured driver is randomly selected from this group. What is the probability that the randomly selected insured driver will have no auto accident in the next policy year? What is the probability that the randomly selected insured driver will have more than 1 auto accident in the next policy year? What is the variance of the number of auto accidents for the randomly selected insured drivers in the next policy year?

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 Practice Problem 3-F The number of TV sets of a particular brand sold in a given week at an electronic store has a Poisson distribution with mean 4. Determine the probability that the store will sell more than 4 TV sets next week. Determine the minimum number of TV sets that the manager should order for the next week so that the probability of having more sales than available TV sets is less than 0.10.

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 Practice Problem 3-G The number of vacant rooms in a given night in a certain hotel follows a Poisson distribution with mean 1.75. Three travelers without reservation walk into the hotel one night. Assume that they do not know each other. Determine the probability that rooms are available for all three travelers. Given that rooms are available for all three travelers, determine the probability that the hotel will still be able to accommodate three more travelers without reservation who also do not know each other.

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 Practice Problem 3-H Cars running the red light arrive at a busy intersection according to a Poisson process with the rate of 0.5 per hour. What is the probability that there will be at most 4 cars running the red light in a 5-hour period? After a period of having no activities in running red light, what is the probability that it will take more than 90 minutes to see another car running the red light? After a period of having no activities in running red light, what is the probability that it will take more than 90 minutes to see two cars running the red light?

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 Practice Problem 3-I Consider a roulette wheel consisting of 38 numbers – 1 through 36, and 0 and 00. A player always makes bets on one of the numbers 1 through 12. Determine the probability that the player will lose his first 5 bets. Determine the probability that the first win of the player will occur on the 5th bet. Determine the probability that the first win of the player will occur no later than the 5th bet.

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 Practice Problem 3-J Suppose that roughly 10% of the adult population have type II diabetes. A researcher wishes to find 3 adult patients who are diabetic. Suppose that the researcher evaluate one patient at a time until finding three diabetic patients. What is the probability that the third diabetic patient is found after evaluating 10 or 11 patients?

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 Practice Problem 3-K For any high risk insured driver, the number of auto accidents in a year has a negative binomial distribution with mean 1.6 and variance 2.88. One such insured driver is selected at random and observed for one year. What is the probability that the insured driver will have more than one accident?

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 Practice Problem 3-L A discrete probability distribution has the following probability function. $\displaystyle P(X=k)=\frac{(k+1) (k+2)}{2} \ \biggl(\frac{4}{9} \biggr)^3 \ \biggl(\frac{5}{9} \biggr)^k \ \ \ \ \ k=0,1,2,3,\cdots$ Determine the mean and variance of $X$.

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 Practice Problem 3-M A large pool of insureds is made up of two subgroups – low risk (75% of the pool) and high risk (25% of the pool). The number of claims in a year for each insured can be any non-negative integer 0, 1, 2, 3, … The number of claims in a year for each insured in the low risk group has a negative binomial distribution with mean 0.5 and variance 0.625. The number of claims in a year for each insured in the high risk group has a negative binomial distribution with mean 0.75 and variance 0.9375. If a randomly selected insured from the pool is observed to have one claim in a given year, what is the probability that the insured is a high risk insured?

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 Practice Problem 3-N An American roulette wheel has 38 areas – numbers 1 through 36 and 0 and 00. A player bets on odd numbers (1, 3, 5, 7, …, 35). He leaves the game when he wins 5 bets. What is the expected number of bets the player will lose before winning 5 bets? What is the probability that the player will lose 5 bets before leaving the game?

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3-A
• 0.171367
• 0.2247123
• $10 e^{-1}=3.67879$
3-B
• 0.388889698
• 0.081670443
• 0.010882596
• 0.000997406
3-C
• 0.127199186
• 0.754604255
3-D
• 0.640545556
• 0.348149413
3-E
• 0.43425
• 0.16795
• 0.6264
3-F
• $\displaystyle 1-\frac{103}{3} e^{-4}=0.371163065$
• min is 7 since $P(X>6)=0.11$ and $P(X>7)=0.0511$
3-G
• $\displaystyle 1-4.28125 e^{-1.75}=0.256030305$
• 0.035673762
3-H
• 0.891178019
• $\displaystyle e^{-0.75}=0.472366553$
• $\displaystyle 1.75 e^{-0.75}=0.826641467$
3-I
• $\displaystyle (13/19)^5=0.1499507895$
• $\displaystyle (6/19) (13/19)^4=0.0692$
• $\displaystyle 1-(13/19)^5=0.85$
3-J
• 0.036589713
3-K
• $\displaystyle \frac{304}{729}=0.417$
3-L
• 3.75
• 8.4375
3-M
• $\displaystyle \frac{0.0768}{0.2688}=0.2857$
3-M
• $\displaystyle \frac{50}{9}=5.56$
• 0.1213520403

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$\copyright$ 2018 – Dan Ma

## Practice Problem Set 2 – Poisson and Gamma

This post presents exercises on gamma distribution and Poisson distribution, reinforcing the concepts discussed in this blog post in a companion blog and blog posts in another blog. Because the shape parameter of the gamma distribution in the following problems is a positive integer, the calculation of probabilities for the gamma distribution is based on Poisson distribution.

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 Practice Problem 2-A Suppose that $X$ is the useful working life (in years) of a brand new industrial machine. The following is the probability density function of $X$. $\displaystyle f(t)=\frac{1}{24} \ \biggl(\frac{1}{5}\biggr)^5 \ t^4 \ e^{-\frac{1}{5} \ t} \ \ \ \ \ \ t>0$ A manufacturing plant has just purchased such a new machine. Determine the probability that the machine will be in operation for the next 20 years.

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 Practice Problem 2-B The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function. $\displaystyle F(x)=1-e^{-0.2 x}-0.2 \ x \ e^{-0.2 x}-0.02 \ x^2 \ e^{-0.2 x} \ \ \ \ \ \ \ 0 In a year in which the annual rainfall is above 20 inches, determine the probability that the annual rainfall is above 30 inches.

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 Practice Problem 2-C The annual rainfall (in inches) in Western Colorado is modeled by a distribution with the following cumulative distribution function. $\displaystyle F(x)=1-e^{-0.2 x}-0.2 \ x \ e^{-0.2 x}-0.02 \ x^2 \ e^{-0.2 x} \ \ \ \ \ \ \ 0 Determine the mean and the variance of the annual rainfall in this region.

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 Practice Problem 2-D The repair time (in hours) for an industrial machine has a gamma distribution with mean 1.5 and variance 0.75. Determine the probability that a repair time exceeds 2 hours. Determine the probability that a repair time is at least 5 hours given that it already exceeds 2 hours.

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 Practice Problem 2-E Customers arriving at a jewelry store according to a Poisson process with an average rate of 2.5 per hours. The store opens its door at 9 AM. What is the probability that the first customer arrives at the store before 11 AM? What is the probability that the first two customers arrive at the store before 11 AM? What is the probability that the first three customers arrive at the store before 11 AM? What is the probability that the first five customers arrive at the store before 11 AM?

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 Practice Problem 2-F In a certain city, telephone calls to 911 emergency response system arrive on the average of two every 3 minutes. Suppose that the arrivals of 911 calls are modeled by a Poisson process. What is the probability of four or more calls arriving in a 5-minute period? A call to the 911 system just ended. What is the probability that the wait time for the next 6 calls is more than 10 minutes?

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 Practice Problem 2-G Customers arrive at a store at an average rate of 30 per hour according to a Poisson process. Determine the probability that at least 5 customers arrive at the store in the first 10 minutes after opening on a given day. Determine the probability that, after opening, it will take more than 15 minutes for the 6th customer to arrive at the store.

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 Practice Problem 2-H Cars arrive at a highway tollbooth at an average rate of 6 cars every 10 minutes according to a Poisson process. Determine the probability that the toll collector will have to wait longer than 20 minutes before collecting the seventh toll. A toll collector just starts his shift. Determine the median time (in minutes) until he collects the first toll.

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 Practice Problem 2-I The number of claims in a year for an insured from a large group of insureds is modeled by the following model. $\displaystyle P(X=x \lvert \lambda)=\frac{e^{-\lambda} \lambda^x}{x!} \ \ \ \ \ x=0,1,2,3,\cdots$ The parameter $\lambda$ varies from insured to insured. However, it is known that $\lambda$ is modeled by the following density function. $\displaystyle g(\lambda)=32 \ \lambda^2 \ e^{-4 \lambda} \ \ \ \ \ \ \lambda>0$ An insured is randomly selected from the large group of insureds. Determine the mean and the variance of the number of claims for this insured in the next year.

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 Practice Problem 2-J Suppose that the number of accidents per year per driver in a large group of insured drivers follows a Poisson distribution with mean $\lambda$. The parameter $\lambda$ follows a gamma distribution with mean 0.9 and variance 0.27. Given that a randomly selected insured has at least one claim, determine the probability that the insured has more than one claim.

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 Practice Problem 2-K Customers arrive at a shop according to a Poisson process. The waiting time (in minutes) until the 5th customer is modeled by the following density function. $\displaystyle f(t)=324 \ t^4 \ e^{-6 \ t} \ \ \ \ \ \ t>0$ Determine mean and variance of the time until the 6th customer after the opening of the shop on a given day. Determine the probability that the wait for the 7th customer is longer than 2 minutes.

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2-A
• $\displaystyle \frac{103}{3} e^{-4}=0.6288$
2-B
• $\displaystyle \frac{25}{13} e^{-2}=0.2603$
2-C
• mean = 15
• variance = 75
2-D
• $\displaystyle P(X>2)=13 e^{-4}$
• $\displaystyle P(X>5 \lvert X>2)=\frac{61}{13} \ e^{-6}=0.01163$
2-E
• $\displaystyle 1-e^{-5}=0.99326$
• $\displaystyle 1-6 e^{-5}=0.95957$
• $\displaystyle 1-18.5 e^{-5}=0.87535$
• $\displaystyle 1-\frac{1569}{24} e^{-5}=0.55951$
2-F
• $\displaystyle 1-\frac{1301}{81} e^{-10/3}=0.4270$
• $\displaystyle 1-\frac{197789}{729} e^{-20/3}=0.6547$
2-G
• $\displaystyle 1-\frac{1569}{24} e^{-5}=0.5595$
• $\displaystyle \frac{52383.28125}{120} e^{-7.5}=0.241436$
2-H
• $\displaystyle 7457.8 e^{-12}=0.04582$
• $\displaystyle \frac{\text{ln}(0.5)}{-0.6}=1.15525$ min
2-I
• mean = 0.75
• variance = 0.9375
2-J
• $\displaystyle \frac{0.6561}{1.3^4}=0.22972$
2-K
• mean = 1, variance = 1/6
• $\displaystyle 7457.8 e^{-12}=0.04582$

Daniel Ma mathematics

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$\copyright$ 2018 – Dan Ma

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