Probability Distributions: a Few Practice Problems

Quantitative Methods – I

Probability Distributions: A few Practice Problems

You are advised to attempt the following problems after going through text book solved examples and attempting problems given in text book and Schaum Series.

Prob. 1

After extensive lab testing of some electrical component, the manufacturer has determined that the average number of such components that fail prior to 1000 hours of operation is two. A buyer of a large number of these items has observed recently that five of them have failed inside 1000 hours. If the number of items that fail within 1000 hours is a Poisson random variable, is there sufficient evidence to doubt the manufacturer’s conclusion.

Prob. 2

A buyer of large quantities of integrated circuits has adopted an acceptance plan that calls for inspection of a sample of 100 chips randomly selected from a very large lot. If the buyer finds no more than two defective chips in the sample, the entire lot is accepted; otherwise the lot is rejected. If a lot containing 1 percent defectives is shipped to the buyer, what is the probability that it will be accepted?

Prob. 3

The management of a restaurant that operates on reservations only knows from experience that 15% of persons making table reservations will not show up. If the restaurant accepts 25 table reservations but has only 20 tables, what is the probability that all who show up will be accommodated?

Prob. 4

The average number of serious accidents at a large industrial plant has been 10 per year, so management has instituted what it believes is an effective plan to reduce the number. In the year after the plan was implemented, four serious accidents occurred. How likely is the occurrence of four of fewer accidents per year if the average rate is still 10 per year? Thus, would one conclude that there might have been a reduction in the average number of accidents per year?

Prob. 5

A recent engineering graduate wishes to take the CAT exam. If the number of times the test is taken constitutes a set of independent events with probability of selection at any test equal to 0.6, what is the probability that no more than 4 attempts will be necessary to get selected. Are the assumptions of independence and constant probability likely to prevail here?

Prob. 6

Let X ~ N(50,10).

a) Determine the following probabilities:

i) P(X < 40) ii) P(X > 35) iii) P(40 < X < 75)

b) Find the values of x corresponding to following probabilities:

i) P(X < x) = 0.125 ii) P(X > x) = 0.386

Prob. 7

Let X ~ N(m,s). If the quantiles x0.4 = 50 and x0.8 = 100, determine the mean and variance of X.

Prob. 8

If X is uniformly distributed with E(X) = 10 and Var(X) = 12, what is the minimum and maximum possible value of X?

Prob. 9

A manufacturer of automobile mufflers wishes to guarantee her mufflers for the life of the automobile. The manufacturer assumes the life of her mufflers is a normally distributed random variable with average life of 3 years and a standard deviation of 6 months. If the unit replacement cost is $50, what whould be the total replacement cost for the first two years if 1,00,000 such mufflers are installed?

Prob. 10

The monthly demand for product A is normally distributed with mean 200 units and standard deviation 40 units. The demand for another product, B, is also normally distributed with mean 500 units and standard deviation 80 units. If a seller of these products stocks 280 units of A and 650 units of B at the beginning of a month, what is the probability that the seller will experience a stockout for both products during the month? You may assume independence.

Prob. 11

The weight of cereal in a container is well approximated by a normal distribution with mean of 600 grams. The filling process is to be designed so that the weight of no more than one container out of 100 will be outside the range 590-610 grams. What is the maximum value for the standard deviation necessary to meet this requirement?

Prob. 12

Which probability distribution would you use to find binomial probabilities the following situations: binomial, Poisson, or normal?

(a) 112 trials, probability of success 0.06.

(b) 15 trials, probabilities of success 0.4.

(c ) 650 trials, probability of success 0.02.

(d) 59 trials, probability of success 0.1.

Prob. 13

Reginald Dunfey, president of British World Airlines, is fiercely proud of his company's on-time percentage; only 2 percent of all BWA flights arrive more than 10 minutes early or late. In his upcoming speech to the BWA board of directors, Mr. Dunfey wants to include the probability that none of the 200 flights scheduled for the following week will be more than 10 minutes early or late. What is the probability? What is the probability that exactly 10 flights will be more than 10 minutes early or late?

Prob. 14

The City Bank of Durham has recently begun a new credit program. Customers meeting certain requirements can obtain a credit card accepted by participating area merchants that carries a discount. Past numbers show that 25 percent of all applicants for this card are rejected. Given that credit acceptance or rejection is a Bernoulli process, out of 15 applicants, what is the probability that

(a) Exactly four will be rejected?

(b) Exactly eight?

(c ) Fewer than three?

(d) More than five?

Prob. 15

Heidi Tanner is the manager of an exclusive shop that sells women's leather clothing and accessories. At the beginning of the fall/winter season, Ms. Tanner must decide how many full-length leather coats to order. These coats cost Ms. Tanner $ 100 each and will sell for $ 200 each. Any coats left over at the end of the season will have to be sold at a 20 percent discount in order to make room for spring/summer inventory. From past experience, Heidi knows that demand for the coats has the following probability distribution:

Number of coats demanded 8 10 12 14 16

Probability 0.10 0.20 0.25 0.30 0.15

She also knows that any leftover coats can be sold at discount.

(a) If Heidi decides to order 14 coats, what is her expected profit.

(b) How would the answer to part (a) change if the leftover coats were sold at a 40 percent discount.

Prob. 16

The purchasing agent in charge of procuring automobiles for the state of Minnesota's interagency motor pool was considering two different models. Both were 4-door, 4-cylinder cars with comparable service warranties. The decision was to choose the automobile that achieved the best mileage per gallon. The state had done some tests of its own, which produced the following results for the two automobiles in question:

Average MPG Standard Deviation

Automobile A 42 4

Automobile B 38 7

The purchasing agent was uncomfortable with the standard deviations, so she set her own decision criterion for the car that would be most likely to get more than 45 miles per gallon.

(a) Using the data provided in combination with the purchasing agent's decision criterion, which car should she choose?

(b) If the purchasing agent's criterion was to reject the automobile that most likely obtained less than 39 mpg, which car should she buy?

Prob. 17

Sensurex Productions, Incorporated, has recently patented and developed an ultrasensitive smoke detector for use in both residential and commercial buildings. Whenever a detectable amount of smoke is in the air, a wailing siren is set off. In recent tests conducted in a 20' x 15' x 8' room, the smoke levels that activated the smoke detector averaged 320 parts per million (ppm) of smoke in the room, with a standard deviation of 25 ppm.

(a) If a cigarette introduces 82 ppm into the atmosphere of a 20' x 15' x 8' room, what is the probability that four people smoking cigarettes simultaneously will set off the alarm?

(b) Three people?

Exponential Distribution

Prob. 18

Let X be exponentially distributed.

a)  What is the probability of a value of X exceeding the mean?

b)  What are the probabilities of X being within one standard deviation of the mean and within two standard deviations of the mean.

Prob. 19

If the life of a component follows exponential distribution, and its reliability of lasting beyond t=55 is 0.4, what is the reliability of component lasting beyond t=100.

Prob. 20

A device has a failure rate of 0.02 per hour. If 500 such devices are used in a plant, what is the probability that no more than 3 devices will fail in one hour?

Prob. 21

The failure time of a component follows exponential distribution with a mean failure time of 5 hours.

a)  What is the probability that the component will last for at least 10 hours?

b)  What is the probability that the component will last at least 20 hours?

c)  If the component has not failed in first 10 hours, what is the probability that will not fail for another 10 hours?