Page 108 - Applied statistics and probability for engineers

86 Chapter 3/Discrete Random Variables and Probability Distributions

3-108. Samples of 20 parts from a metal punching pro- 3-112. Consider the patient data in Example 2-8. Suppose that
cess are selected every hour. Typically, 1% of the parts require ive patients are randomly selected with replacement from the
rework. Let X denote the number of parts in the sample of total for hospital 4. Determine the following probabilities:
20 that require rework. A process problem is suspected if X (a) Exactly one is LWBS. (b) Two or more are LWBS.
exceeds its mean by more than 3 standard deviations. (c) At least one is LWBS.
(a) If the percentage of parts that require rework remains at 3-113. Assume that a Web site changes its content according
1%, what is the probability that X exceeds its mean by to the distribution in Exercise 3-34. Assume that 10 changes
more than 3 standard deviations? are made independently.
(b) If the rework percentage increases to 4%, what is the (a) What is the probability that the change is made in less than
probability that X exceeds 1? 4 days in 7 of the 10 updates?
(c) If the rework percentage increases to 4%, what is the (b) What is the probability that the change is made in less than
probability that X exceeds 1 in at least one of the next 4 days in 2 or fewer of the 10 updates?
ive hours of samples? (c) What is the probability that at least one change is made in
3-109. Because all airline passengers do not show up less than 4 days?
for their reserved seat, an airline sells 125 tickets for a light (d) What is the expected number of the 10 updates that occur
that holds only 120 passengers. The probability that a pas- in less than 4 days?
senger does not show up is 0.10, and the passengers behave 3-114. Consider the endothermic reactions in Exercise 3-32.
independently. A total of 20 independent reactions are to be conducted.
(a) What is the probability that every passenger who shows up (a) What is the probability that exactly 12 reactions result in a
can take the light? inal temperature less than 272 K?
(b) What is the probability that the light departs with empty (b) What is the probability that at least 19 reactions result in a
seats? inal temperature less than 272 K?
3-110. This exercise illustrates that poor quality can (c) What is the probability that at least 18 reactions result in a
affect schedules and costs. A manufacturing process has 100 inal temperature less than 272 K?
customer orders to ill. Each order requires one component (d) What is the expected number of reactions that result in a
part that is purchased from a supplier. However, typically, 2% inal temperature of less than 272 K?
of the components are identiied as defective, and the compo- 3-115. The probability that a visitor to a Web site provides
nents can be assumed to be independent. contact data for additional information is 0.01. Assume that
(a) If the manufacturer stocks 100 components, what is the prob- 1000 visitors to the site behave independently. Determine the
ability that the 100 orders can be illed without reordering following probabilities:
components? (a) No visitor provides contact data.
(b) If the manufacturer stocks 102 components, what is the prob- (b) Exactly 10 visitors provide contact data.
ability that the 100 orders can be illed without reordering (c) More than 3 visitors provide contact data.
components? 3-116. Consider the circuit in Example 2-34. Assume that devices
(c) If the manufacturer stocks 105 components, what is the prob- fail independently. What is the probability mass function of the
ability that the 100 orders can be illed without reordering number of device failures? Explain why a binomial distribution
components? does not apply to the number of device failures in Example 2-32.
3-111. Consider the lengths of stay at a hospital’s emer- 3-117. Consider the time to recharge the lash in cell-phone cam-
gency department in Exercise 3-33. Assume that ive persons eras as in Example 3-2. Assume that the probability that a camera
independently arrive for service. passes the test is 0.8 and the cameras perform independently. What
(a) What is the probability that the length of stay of exactly is the smallest sample size needed so that the probability of at least
one person is less than or equal to 4 hours? one camera failing is at least 95%?
(b) What is the probability that exactly two people wait more 3-118. Consider the patient data in Example 2-8. Suppose that
than 4 hours? patients are randomly selected with replacement from the total
(c) What is the probability that at least one person waits more for hospital 4. What is the smallest sample size needed so that
than 4 hours? the probability is at least 90% that at least one patient is LWBS?

3-7 Geometric and Negative Binomial Distributions

3-7.1 GEOMETRIC DISTRIBUTION
Consider a random experiment that is closely related to the one used in the deinition of a
binomial distribution. Again, assume a series of Bernoulli trials (independent trials with
constant probability p of a success on each trial). However, instead of a ixed number of
trials, trials are conducted until a success is obtained. Let the random variable X denote the