## 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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