Suppose {A,B} ci |C{A,B} ci |C and {A,B} ci |Cc{A,B} ci |Cc, P(C)=0.7P(C)=0.7, and

Show whether or not the pair {A,B}{A,B} is independent.

Suppose {A1,A2,A3} ci |C{A1,A2,A3} ci |C and ci |Cc ci |Cc, with P(C)=0.4P(C)=0.4, and

Determine the posterior odds P(C|A1A2cA3)/P(Cc|A1A2cA3)P(C|A1A2cA3)/P(Cc|A1A2cA3).

Five world class sprinters are entered in a 200 meter dash. Each has a good chance to break the current track record. There is a thirty percent chance a late cold front will move in, bringing conditions that adversely affect the runners. Otherwise, conditions are expected to be favorable for an outstanding race. Their respective probabilities of breaking the record are:

The performances are (conditionally) independent, given good weather, and also, given poor weather. What is the probability that three or more will break the track record?

Hint. If B₃ is the event of three or more, P(B3)=P(B3|W)P(W)+P(B3|Wc)P(Wc)P(B3)=P(B3|W)P(W)+P(B3|Wc)P(Wc).

A device has five sensors connected to an alarm system. The alarm is given if three or more of the sensors trigger a switch. If a dangerous condition is present, each of the switches has high (but not unit) probability of activating; if the dangerous condition does not exist, each of the switches has low (but not zero) probability of activating (falsely). Suppose D=D= the event of the dangerous condition and A=A= the event the alarm is activated. Proper operation consists of AD⋁AcDcAD⋁AcDc. Suppose Ei=Ei= the event the ith unit is activated. Since the switches operate independently, we suppose

Assume the conditional probabilities of the E₁, given D, are 0.91, 0.93, 0.96, 0.87, 0.97, and given D^c, are 0.03, 0.02, 0.07, 0.04, 0.01, respectively. If P(D)=0.02P(D)=0.02, what is the probability the alarm system acts properly? Suggestion. Use the conditional independence and the procedure ckn.

Seven students plan to complete a term paper over the Thanksgiving recess. They work independently; however, the likelihood of completion depends upon the weather. If the weather is very pleasant, they are more likely to engage in outdoor activities and put off work on the paper. Let E_i be the event the ith student completes his or her paper, A_k be the event that k or more complete during the recess, and W be the event the weather is highly conducive to outdoor activity. It is reasonable to suppose {Ei:1≤i≤7} ci |W{Ei:1≤i≤7} ci |W and ci |Wc ci |Wc. Suppose

respectively, and P(W)=0.8P(W)=0.8. Determine the probability P(A4)P(A4) that four our more complete their papers and P(A5)P(A5) that five or more finish.

A manufacturer claims to have improved the reliability of his product. Formerly, the product had probability 0.65 of operating 1000 hours without failure. The manufacturer claims this probability is now 0.80. A sample of size 20 is tested. Determine the odds favoring the new probability for various numbers of surviving units under the assumption the prior odds are 1 to 1. How many survivors would be required to make the claim creditable?

A real estate agent in a neighborhood heavily populated by affluent professional persons is working with a customer. The agent is trying to assess the likelihood the customer will actually buy. His experience indicates the following: if H is the event the customer buys, S is the event the customer is a professional with good income, and E is the event the customer drives a prestigious car, then

Since buying a house and owning a prestigious car are not related for a given owner, it seems reasonable to suppose P(E|HS)=P(E|HcS)P(E|HS)=P(E|HcS) and P(E|HSc)=P(E|HcSc)P(E|HSc)=P(E|HcSc). The customer drives a Cadillac. What are the odds he will buy a house?

In deciding whether or not to drill an oil well in a certain location, a company undertakes a geophysical survey. On the basis of past experience, the decision makers feel the odds are about four to one favoring success. Various other probabilities can be assigned on the basis of past experience. Let

The initial, or prior, odds are P(H)/P(Hc)=4P(H)/P(Hc)=4. Previous experience indicates

Make reasonable assumptions based on the fact that the result of the geophysical survey depends upon the geological formations and not on the presence or absence of oil. The result of the survey is favorable. Determine the posterior odds P(H|E)/P(Hc|E)P(H|E)/P(Hc|E).

A software firm is planning to deliver a custom package. Past experience indicates the odds are at least four to one that it will pass customer acceptance tests. As a check, the program is subjected to two different benchmark runs. Both are successful. Given the following data, what are the odds favoring successful operation in practice? Let

Under the usual conditions, we may assume {H,E1,E2} ci |S{H,E1,E2} ci |S and ci |Sc ci |Sc. Reliability data show

Determine the posterior odds P(H|E1E2)/P(Hc|E1E2)P(H|E1E2)/P(Hc|E1E2).

A research group is contemplating purchase of a new software package to perform some specialized calculations. The systems manager decides to do two sets of diagnostic tests for significant bugs that might hamper operation in the intended application. The tests are carried out in an operationally independent manner. The following analysis of the results is made.

Since the tests are for the presence of bugs, and are operationally independent, it seems reasonable to assume {H,E1,E2} ci |S{H,E1,E2} ci |S and {H,E1,E2} ci |Sc{H,E1,E2} ci |Sc. Because of the reliability of the software company, the manager thinks P(S)=0.85P(S)=0.85. Also, experience suggests

Determine the posterior odds favoring H if results of both diagnostic tests are satisfactory.

A company is considering a new product now undergoing field testing. Let

The company assumes P(S)=0.9P(S)=0.9 and the conditional probabilities

Since the testing of the merchandise is not affected by market success or failure, it seems reasonable to suppose {H,E} ci |S{H,E} ci |S and ci |Sc ci |Sc. The field tests are favorable. Determine P(H|E)/P(Hc|E)P(H|E)/P(Hc|E).

Martha is wondering if she will get a five percent annual raise at the end of the fiscal year. She understands this is more likely if the company's net profits increase by ten percent or more. These will be influenced by company sales volume. Let

Since the prospect of a raise depends upon profits, not directly on sales, she supposes {H,E} ci |S{H,E} ci |S and {H,E} ci |Sc{H,E} ci |Sc. She thinks the prior odds favoring suitable profit increase is about three to one. Also, it seems reasonable to suppose

End of the year records show that sales increased by eighteen percent. What is the probability Martha will get her raise?

A physician thinks the odds are about 2 to 1 that a patient has a certain disease. He seeks the “independent” advice of three specialists. Let H be the event the disease is present, and A,B,CA,B,C be the events the respective consultants agree this is the case. The physician decides to go with the majority. Since the advisers act in an operationally independent manner, it seems reasonable to suppose {A,B,C} ci |H{A,B,C} ci |H and ci |Hc ci |Hc. Experience indicates

What is the probability of the right decision (i.e., he treats the disease if two or more think it is present, and does not if two or more think the disease is not present)?

A software company has developed a new computer game designed to appeal to teenagers and young adults. It is felt that there is good probability it will appeal to college students, and that if it appeals to college students it will appeal to a general youth market. To check the likelihood of appeal to college students, it is decided to test first by a sales campaign at Rice and University of Texas, Austin. The following analysis of the situation is made.

Since the tests are for the reception are at two separate universities and are operationally independent, it seems reasonable to assume {H,E1,E2} ci |S{H,E1,E2} ci |S and {H,E1,E2} ci |Sc{H,E1,E2} ci |Sc. Because of its previous experience in game sales, the managers think P(S)=0.80P(S)=0.80. Also, experience suggests

Determine the posterior odds favoring H if sales results are satisfactory at both schools.