Page 119 - Applied statistics and probability for engineers

Section 3-8/Hypergeometric Distribution 97

(a) What is the probability mass function of the number of cells (c) Suppose that on the irst day of evaluation, 2 of the blades
in the sample that can replicate? are dull; on the second day of evaluation, 6 are dull; and
(b) What are the mean and variance of the number of cells in on the third day of evaluation, 10 are dull. What is the
the sample that can replicate? probability that the assembly is not replaced until the third
(c) What is the probability that at least one of the selected cells day of evaluation? [Hint: Assume that the daily decisions
cannot replicate? are independent. However, the probability of replacement
3-145. A research study uses 800 men under the age of 55. changes every day.]
Suppose that 30% carry a marker on the male chromosome that 3-1 50. Calculate the inite population corrections
indicates an increased risk for high blood pressure. (a) For Exercises 3-141 and 3-142, for which exercise should
(a) If 10 men are selected randomly and tested for the marker, the binomial approximation to the distribution of X be
what is the probability that exactly 1 man has the marker? better?
(
(
(b) If 10 men are selected randomly and tested for the marker, (b) For Exercise 3-141, calculate P X = ) 1 and P X = ) 4 ,
what is the probability that more than 1 has the marker? assuming that X has a binomial distribution, and compare
3-146. Printed circuit cards are placed in a functional test these results to results derived from the hypergeometric
after being populated with semiconductor chips. A lot contains distribution.
(
(
140 cards, and 20 are selected without replacement for func- (c) For Exercise 3-142, calculate P X = ) 1 and P X = ) 4 ,
tional testing. assuming that X has a binomial distribution, and compare
(a) If 20 cards are defective, what is the probability that at least these results to the results derived from the hypergeometric
1 defective card is in the sample? distribution.
(b) If 5 cards are defective, what is the probability that at least (d) Use the binomial approximation to the hypergeometric dis-
1 defective card appears in the sample? tribution to approximate the probabilities in Exercise 3-146.
3-147. The analysis of results from a leaf transmutation What is the inite population correction in this exercise?
experiment (turning a leaf into a petal) is summarized by the 3-151. Consider the visits that result in leave without being seen
type of transformation completed: (LWBS) at an emergency department in Example 2-8. Assume that
Total Textural four visits that result in LWBS are to be randomly selected (with-
Transformation out replacement) for a follow-up interview.
Yes No (a) What is the probability that all selected visits are from hos-
Total Color Yes 243 26 pital 4?
Transformation No 13 18 (b) What is the probability that no selected visits are from hos-
pital 4?
A naturalist randomly selects three leaves from this set without
(c) What is the probability that all selected visits are from the
replacement. Determine the following probabilities.
same hospital?
(a) Exactly one has undergone both types of transformations.
(b) At least one has undergone both transformations. 3-152. Consider the nonfailed wells in Exercises 3-35. Assume
(c) Exactly one has undergone one but not both transformations. that four wells are selected randomly (without replacement) for
(d) At least one has undergone at least one transformation. inspection.
3-148. A state runs a lottery in which six numbers are (a) What is the probability that exactly two are selected from
randomly selected from 40 without replacement. A player the Loch Raven Schist?
chooses six numbers before the state’s sample is selected. (b) What is the probability that one or more is selected from the
(a) What is the probability that the six numbers chosen by a Loch Raven Schist?
player match all six numbers in the state’s sample? (c) What is the expected number selected from the Loch Raven
(b) What is the probability that ive of the six numbers chosen Schist?
by a player appear in the state’s sample? 3-153. Consider the semiconductor wafer data in Table 2-1. Sup-
(c) What is the probability that four of the six numbers chosen pose that 10 wafers are selected randomly (without replacement)
by a player appear in the state’s sample? for an electrical test. Determine the following:
(d) If a player enters one lottery each week, what is the expected (a) Probability that exactly 4 wafers have high contamination.
number of weeks until a player matches all six numbers in (b) Probability that at least 1 is from the center of the sputter-
the state’s sample? ing tool and has high contamination.
3-149. A slitter assembly contains 48 blades. Five blades (c) Probability that exactly 3 have high contamination or are
are selected at random and evaluated each day for sharpness. If from the edge of the sputtering tool.
any dull blade is found, the assembly is replaced with a newly (d) Instead of 10 wafers, what is the minimum number of wafers
sharpened set of blades. that need to be selected so that the probability that at least 1
(a) If 10 of the blades in an assembly are dull, what is the probabil- wafer has high contamination is greater than or equal to 0.9?
ity that the assembly is replaced the irst day it is evaluated? 3-154. Suppose that a healthcare provider selects 20 patients
(b) If 10 of the blades in an assembly are dull, what is the prob- randomly (without replacement) from among 500 to evaluate
ability that the assembly is not replaced until the third day adherence to a medication schedule. Suppose that 10% of the
of evaluation? [Hint: Assume that the daily decisions are 500 patients fail to adhere with the schedule. Determine the
independent, and use the geometric distribution.] following: