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Chapter 3: Q97E (page 203)
Consider two events A and B, with,,and
a.Are A and B mutually exclusive?
b.Are A and B independent?
a. The events are mutually exclusive.
b. The events are not independent.
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Patient medical instruction sheets. Physicians and pharmacists sometimes fail to inform patients adequately about the proper application of prescription drugs and about the precautions to take in order to avoid potential side effects. One method of increasing patients鈥� awareness of the problem is for physicians to provide patient medication instruction (PMI) sheets. The American Medical Association, however, has found that only 20% of the doctors who prescribe drugs frequently distribute PMI sheets to their patients. Assume that 20% of all patients receive the PMI sheet with their prescriptions and that 12% receive the PMI sheet and are hospitalized because of a drug-related problem. What is the probability that a person will be hospitalized for a drug-related problem given that the person received the PMI sheet?
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鈥攏amely, 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 probabilityin 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?
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 鈥淥K鈥� or 鈥渘ot OK.鈥� Let Arepresent the event that an 鈥淥K鈥� report is generated in any time period (t).Let Brepresent the event that an 鈥淥K鈥� report is generated in the next time period. Consider the following probabilities:
,, and.
a. Express the event B|Ain 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鈥� Rule to find P(A|B), i.e., the probability that an 鈥淥K鈥� report was generated in one time period(t), given that an 鈥淥K鈥� report is generated in the next time period.
The diagram below describes the sample space of a particular experiment and events A and B .
Simulate the experiment described in Exercise 3.7 using any five identically shaped objects, two of which are one colour and the three another colour. Mix the objects, draw two, record the results, and then replace the objects. Repeat the experiment a large number of times (at least 100). Calculate the proportion of time events A, B, and C occur. How do these proportions compare with the probabilities you calculated in Exercise 3.7? Should these proportions equal the probabilities? Explain.
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