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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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Advertising proposals. The manager of an advertising department has asked her creative team to propose six new ideas for an advertising campaign for a major client. She will choose three of the six proposals to present to the client. The proposals were named A, B, C, D, E, and F, respectively.
a. In how many ways can the manager select the three proposals? List the possibilities.
b. It is unlikely that the manager will randomly select three of the six proposals, but if she does, what is the probability that she will select proposals A, D, and E?
Male nannies. In a survey conducted by the International Nanny Association (INA) and reported on the INA Web site (www.nanny.org), 4,176 nannies were placed in a job in a given year. Only 24 of the nannies placed were men. Find the probability that a randomly selected nanny placed during the last year is a male nanny (a 鈥渕annie鈥�).
Two fair coins are tossed, and the following events are defined:
A: [Observe one head and one tail.]
B: [Observe at least one head.]
a. Define the possible sample points and assign probabilities to each.
b. Draw a Venn diagram for the experiment, showing the sample points and events A and B.
c. Find P(A), P(B) and.
d. Use the formula for conditional probability to find P (A/B)and P (B/A). Verify your answer by inspecting the Venn diagram and using the concept of reduced sample spaces.
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.
Two fair dice are tossed, and the following events are defined:
A: {Sum of the numbers showing is odd.}
B: {Sum of the numbers showing is 9, 11, or 12.}
Are events A and B independent? Why?
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