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.
Practice Problem 3A 
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.

Practice Problem 3B 
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.
See here for the calculation of Powerball winning odds. 
Practice Problem 3C 
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,

Practice Problem 3D 
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.

Practice Problem 3E 
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.

Practice Problem 3F 
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.

Practice Problem 3G 
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.

Practice Problem 3H 
Cars running the red light arrive at a busy intersection according to a Poisson process with the rate of 0.5 per hour.

Practice Problem 3I 
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.

Practice Problem 3J 
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.

Practice Problem 3K 
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? 
Practice Problem 3L 
A discrete probability distribution has the following probability function. Determine the mean and variance of . 
Practice Problem 3M 
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 nonnegative 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? 
Practice Problem 3N 
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.

Problem  ………..Answer 

3A 

3B 

3C 

3D 

3E 

3F 

3G 

3H 

3I 

3J 

3K 

3L 

3M 

3M 

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