Monitoring quality of power equipment. Mechanical Engineering (February 2005) reported on the need for wireless networks to monitor the quality of industrial equipment. For example, consider Eaton Corp., a company that develops distribution products. Eaton estimates that 90% of the electrical switching devices it sells can monitor the quality of the power running through the device. Eaton further estimates that of the buyers of electrical switching devices capable of monitoring quality, 90% do not wire the equipment up for that purpose. Use this information to estimate the probability that an Eaton electrical switching device is capable of monitoring power quality and is wired up for that purpose.
The probability that an Eaton electrical switching device is capable of monitoring power quality and is wired up for that purpose is 0.09.
The formula for probability is .
Now, find the result, then,
Therefore, the probability is 0.09.
Confidence of feedback information for improving quality. In the semiconductor manufacturing industry, a key to improved quality is having confidence in the feedback generated by production equipment. A study of the confidence level of feedback information was published in Engineering Applications of Artificial Intelligence (Vol. 26, 2013). At any point in time during the production process, a report can be generated. The report is classified as either “OK” or “not OK.” Let A represent the event that an “OK” report is generated in any time period (t).Let B represent the event that an “OK” report is generated in the next time period . Consider the following probabilities:
, , and .
a. Express the event B|A in the words of the problem.
b. Express the event B| in the words of the problem.
c. Find .
d. Find .
e. Find .
f. Use the probabilities, parts d and e, to find P(B).
g. Use Bayes’ Rule to find P(A|B), i.e., the probability that an “OK” report was generated in one time period(t), given that an “OK” report is generated in the next time period .
Working on summer vacation. Is summer vacation a break from work? Not according to a Harris Interactive (July 2013) poll of U.S. adults. The poll found that 61% of the respondents work during their summer vacation, 22% do not work while on vacation, and 17% are unemployed. Assuming these percentages apply to the population of U.S. adults, consider the work status during the summer vacation of a randomly selected adult.
a. What is the Probability that the adult works while on summer vacation?
b. What is the Probability that the adult will not work while on summer vacation, either by choice or due to unemployment?
Blood diamonds. According to Global Research News (March 4, 2014), one-fourth of all rough diamonds produced in the world are blood diamonds. (Any diamond that is mined in a war zone—often by children—to finance a warlord’s activity, an insurgency, or an invading army’s effort is considered a blood diamond.) Also, 90% of the world’s rough diamonds are processed in Surat, India, and, of these diamonds one-third are blood diamonds.
a. Find the probability that a rough diamond is not a blood diamond.
b. Find the probability that a rough diamond is processed in Surat and is a blood diamond.
Consider the experiment depicted by the Venn diagram, with the sample space S containing five sample points. The sample points are assigned the following probabilities:
a. Calculate .
b. Suppose we know that event A has occurred, so that the reduced sample space consists of the three sample points in A—namely, E1, E2, and E3. Use the formula for conditional probability to adjust the probabilities of these three sample points for the knowledge that A has occurred [i.e., ]. Verify that the conditional probabilities are in the same proportion to one another as the original sample point probabilities.
c. Calculate the conditional probability in two ways: (1) Add the adjusted (conditional) probabilities of the sample points in the intersection , as these represent the event that B occurs given that A has occurred; (2) use the formula for conditional probability:
Verify that the two methods yield the same result.
d. Are events A and B independent? Why or why not?
Museum management. Refer to the Museum Management and Curatorship (June 2010) study of the criteria used to evaluate museum performance, Exercise 2.14 (p. 74). Recall that the managers of 30 leading museums of contemporary art were asked to provide the performance measure used most often. A summary of the results is reproduced in the table. Performance Measure Number of Museums Total visitors 8 Paying visitors 5 Big shows 6 Funds raised 7 Members 4
Number of Museums
a. If one of the 30 museums is selected at random, what is the probability that the museum uses total visitors or funds raised most often as a performance measure?
b. Consider two museums of contemporary art randomly selected from all such museums. Of interest is whether or not the museums use total visitors or funds raised most often as a performance measure. Use a tree diagram to aid in listing the sample points for this problem.
c. Assign reasonable probabilities to the sample points of part b.
d. Refer to parts b and c. Find the probability that both museums use total visitors or funds raised most often as a performance measure.
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