Problem 1: Three times each day, a quality engineer samples a component from a recently manufactured batch and tests it. Each part is classified as conforming (suitable for its intended use), downgraded (unsuitable for the intended purpose but usable for another purpose), or scrap (not usable). An experiment consists of recording the categories of the three parts tested in a particular day.
a. List the 27 outcomes in the sample space.
b. Let A be the event that all the parts fall into the same category. List the outcomes in A.
c. Let B be the event that there is one part in each category. List the outcomes in B.
d. Let C be the event that at least two parts are conforming. List the outcomes in C.
e. List the outcomes in A ∩ C.
f. List the outcomes in A ∪ B.
g. List the outcomes in A ∩ Cc.
h. List the outcomes in Ac ∩ C.
i. Are events A and C mutually exclusive? Explain.
j. Are events B and C mutually exclusive? Explain.
Problem 2: A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop after the end of a run. Suppose that the main chute deploys with probability 0.99, and that if the main fails to deploy, the backup deploys with probability 0.98.
a. What is the probability that one of the two parachutes deploys?
b. What is the probability that the backup parachutes deploys?
Problem 3: A lot of 1000 components contain 300 that are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective, and let B be the event that the second component drawn is defective.
a. Find P(A).
b. Find P(B|A).
c. Find P(A ∩ B).
d. Find P(Ac ∩ B).
e. Find P(B) .
f. Find P(A|B).
g. Are A and B independent? Is it reasonable to treat A and B as though they were independent? Explain.
Problem 4: Computer chips often contain surface imperfections. For a certain type of computer chip, the probability mass function of the number of defects X is presented in the following table.
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x
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0
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1
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2
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3
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4
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p(x)
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0.4
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0.3
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0.15
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0.10
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0.05
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a. Find P(X ≤ 2).
b. Find P(X > 1).
c. Find μX.
d. Find σx2.
Problem 5: Let X represent the number of tires with low air pressure on a randomly chosen car.
a. Which of the three functions below is a possible probability mass function of X? Explain.
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X
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|
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0
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1
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2
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3
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4
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p1(x)
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0.2
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0.2
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0.3
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0.1
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0.1
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P2(x)
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0.1
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0.3
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0.3
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0.3
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0.2
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p[3(x)
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0.1
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0.2
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0.2
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0.4
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0.1
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b. For the possible probability mass function, compute μX and σx2.
Problem 6: Three components are randomly sampled, one at a time, from a large lot. As each component is selected, it is tested. If it passes the test, a success (S) occurs; if it fails the test, a failure (F) occurs. Assume that 80% of the components in the lot will succeed in passing the test. Let X represent the number of successes among the three sampled components.
a. What are the possible values for X?
b. Find P(X = 3).
c. The event that the first component fails and the next two succeed is denoted by FSS. Find P(FSS)
d. Find P(SFS) and P(SSF).
e. Use the results of parts (c) and (d) to find P(X = 2).
f. Find P(X= 1).
g. Find P(X = 0).
h. Find μx.
i. Find σx2.
j. Let Y represent the number of successes if four components are sampled. Find P(Y = 3).
Problem 7: Let X ∼ Bin(9, 0.4). Find
a. P(X > 6)
b. P(X ≥ 2)
c. P(2 ≤ X < 5)
d. P(2 < X ≤ 5)
e. P(X = 0)
f. P(X = 7)
g. μx
h. σx2.
Problem 8: A quality engineer takes a random sample of 100 steel rods from a day's production, and finds that 92 of them meet specifications.
a. Estimate the proportion of that day's production that meets specifications, and find the uncertainty in the estimate.
b. Estimate the number of rods that must be sampled to reduce the uncertainty to 1%.
Problem 9: A data center contains 1000 computer servers. Each server has probability 0.003 of failing on a given day.
a. What is the probability that exactly two servers fail?
b. What is the probability that fewer than 998 servers function?
c. What is the mean number of servers that fail?
d. What is the standard deviation of the number of servers that fail?
Problem 10: The number of cars arriving at a given intersection follows a Poisson distribution with a mean rate of 4 per second.
a. What is the probability that 3 cars arrive in a given second?
b. What is the probability that 8 cars arrive in three seconds?
c. What is the probability that more than 3 cars arrive in a period of two seconds?