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5. DISCRETE PROBABILITY DISTRIBUTIONS

Q1) In a certain city district the need for money to buy drugs is stated as the: reason for 75% of all

thefts. Find the probability that among the next 5 theft cases reported in this district,

a. Exactly 2 resulted from the need for money to buy drugs.

b. At most 3 resulted from the need for money to buy.

Q2) In testing a certain kind of truck tire over a rugged terrain, it is found that 25% of the trucks

fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability

that

a. From 3 to 6 have blowouts.

b. Fewer than 4 have blowouts.

c. More than 5 have blowouts.

Q3) The probability that a patient recovers from a delicate heart operation is 0.9. What is the

probability that exactly 5 of the next 7 patients having this operation survive?

Q4) It is known that 60% of mice inoculated with a serum are protected from a certain disease. If

5 mice are inoculated, find the probability that

a. none contracts the disease.

b. fewer than 2 contract the disease.

c. more than 3 contract the disease.

Q5) In a study of brand recognition, 95% of consumers recognized Coke. The company randomly

selects 4 consumers for a taste test. Let X be the number of consumers who recognize Coke.

a. Write out the PMF table for this. b. Find the probability that among the 4 consumers, 2 or more will recognize Coke. c. Find the expected number of consumers who will recognize Coke. d. Find the variance for the number of consumers who will recognize Coke

Q6) Three people toss a fair coin and the odd man pays for coffee. If the coins all turn up the

same, they are tossed again. Find the probability that fewer than 4 tosses are needed.

Q7) According to a study published by a group of University of Massachusetts sociologists, about

two thirds of the 20 million persons in this country who take Valium are women. Assuming this

figure to be a valid estimate, find the probability that on a given day the fifth prescription written

by a doctor for Valium is

a. The first prescribing Valium for a woman.

b. The third prescribing Valium for a woman.

Q8) The probability that a student passes the written test for a private pilot's license is 0.7. Find

the probability that the student will pass the test

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a. On the third try.

b. Before the fourth try. (u can add after, between two points).

Q9) From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective

missiles that will not fire, what is the probability that

a. All 4 will fire?

b. At most 2 will not. fire?

Q10) A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for

the probability distribution of the random variable X representing

the number of doctors on the committee. Find P(2 ≤ X ≤ 3)

Q11) A manufacturing company uses an acceptance scheme on production items before they are

shipped. The plan is a two-stage one. Boxes of 25 are readied for shipment and a sample of 3 is

tested for defectives. If any defectives are found, the entire box is sent back

for 100% screening. If no defectives are found, the box is shipped.

a. What is the probability that a box containing 3 defectives will be shipped?

b. What is the probability that a box containing only 1 defective will be sent back for screening?

Q12) On average a certain intersection results in 3 traffic accidents per month.

For any given month at this intersection. What is the probability that:

a. Exactly 5 accidents will occur?

b. Less than 3 accidents will occur?

c. At least 2 accidents will occur?

For any given year at this intersection. What is the probability that:

a. Exactly 5 accidents will occur?

b. Less than 3 accidents will occur?

c. At least 2 accidents will occur?

Q13) A secretary makes 2 errors per page, on average. What is the probability that on the next

page he or she will make

a. 4 or more errors?

b. No errors?

Q14) A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find

the probability that for a given year that area will be hit by

a. Fewer than 4 hurricanes;

b. Anywhere from 6 to 8 hurricanes.

c. Find the probability that for a given 3 months that area will be hit by fewer than 4 hurricanes.

Q15) When a die is tossed once, each element of the sample space occurs with probability 1/6.

Therefore we have a uniform distribution.

Find:

a.𝑃(1 ≤ 𝑋 < 4) b.𝑃(3 < 𝑋 < 6)

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c.𝑃(𝑋 < 3) d. Find also the mean and variance.

Q16) X has is uniformly distributed on the set {1,2,3,…,N}, and

Y is uniformly distributed on the set {a,a+k,a+2k,…,b}, then find

a. P(X) and P(Y)

b. M(t) for X and for Y

c. E(X) and E(Y)

d. V(X) and Var(Y)

Q17) Suppose our class passed (C or better) the last exam with probability 0.75.

a. Find the probability that someone passes the exam. b. Find the mean value of the random variable c. Find the standard deviation value of the random variable d. Find the moment generating function of the random variable

Q18) 20% from a population have a particular disease. In testing process for infection by this

disease.

a. Find the probability that someone infected by this disease. b. Find the mean value of the random variable c. Find the standard deviation value of the random variable d. Find the moment generating function of the random variable

Q19) Suppose X has a geometric distribution with p=0.8. Compute the probability of the following

events.

Q20) If the probability is 0.75 that an application for a driver's license will pass the road test on

any given try, what is the probability that an application will finally pass the test on the fourth try

Q21) Suppose that 30% of the application for a certain industrial job have advanced training in

computer programming. Application are interviewed sequentially and are selected at random from

the pool. Find the probability that the first application having advanced in programming is found

on the fifth interview.

Q22) Let X be uniformly distributed on 0,1,…,99. Calculate

a. 𝑃(𝑋 ≥ 25).

b. 𝑃(2.6 < 𝑋 < 12.2).

c. 𝑃(8 < 𝑋 ≤ 10 𝑜𝑟 2 < 𝑋 ≤ 32).

d. 𝑃(25 ≤ 𝑋 ≤ 30).

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Q23) If the probability is 0.40 that a child exposed to a certain contagious disease will catch it,

what is the probability that the tenth child exposed to the disease will be the third to catch it.

Q24) In an assembly process, the finished items are inspected by a vision sensor, the image data

is processed , and a determination is made by computer as to whether or not a unit is satisfactory.

If it is assumed that 2% of the units will be rejected, then what is the probability that the thirtieth

unit observed will be second rejected unit?

Q25) If 2 balls are randomly drawn from a bowl containing 6 white and 5 black balls, what is the

probability that one of the drawn balls is white and the other black?

Q26) Of 10 girls in a class, 3 have blue eyes. If two of the girls are chosen at random, what is the

probability that

a. Both have blue eyes.

b. Neither have blue eyes.

c. At least one has blue eyes.

Q27) A company installs new central heating furnaces, and has found that for 15% of all

installations a return visit is needed to make some modifications. Six installations were made in a

particular week. Assume independence of outcomes for these installations.

a. What is the probability that a return visit was needed in all of these cases?

b. What is the probability that a return visit was needed in none of these cases?

c. What is the probability that a return visit was needed in more than one of these cases?

Q28) A fair die is rolled 4 times. Find

a. The probability of obtaining exactly one 6.

b.The probability of obtaining no 6.

c.The probability of obtaining at least one 6.

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Q29) In a study of a drug -induced anaphylaxis among patients taking rocuronium bromide as part

of their anesthesia, Laake and Rottingen found that the occurrence of anaphylaxis followed a

Poisson model with =12 incidents per year in Norway .Find

a. The probability that in the next year, among patients receiving rocuronium, exactly three will

experience anaphylaxis?

b. The probability that less than two patients receiving rocuronium, in the next year will experience

anaphylaxis?

c. The probability that more than two patients receiving rocuronium, in the next two years will

experience anaphylaxis?

d. The expected value of patients receiving rocuronium, in the next 6 months who will experience

anaphylaxis.

e. The variance of patients receiving rocuronium, in the next year who will experience anaphylaxis.

f. The standard deviation of patients receiving rocuronium, in the next year who will experience

anaphylaxis.

Q30) If the probability that an individual will suffer a bad reaction from injection of a given serum

is 0.001, determine the probability that out of 2000 individuals, (a) exactly 3, (b) more than 2,

individuals will suffer.

Q31) Suppose 2% of the items made by a factory are defective. Find the probability that there are

3 defective items in a sample of 100 items.

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