5.3 Problems on Conditional Independence

Events made independent by conditioning on a common cause, with Bayesian calculations weighing evidence for competing hypotheses.

Applied Probability, 5.3 Problems on Conditional Independence

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16 exercises

5.3 Exercise 1

Suppose $\left\{A , B\right\} ci \left|\right. C$ and $\left\{A , B\right\} ci \left|\right. C^{c}$ , $P \left(C\right) = 0 . 7$ , and

P \left(\right. A \left|\right. C \left.\right) = 0 . 4 P \left(\right. B \left|\right. C \left.\right) = 0 . 6 P \left(\right. A \left|\right. C^{c} \left.\right) = 0 . 3 P \left(\right. B \left|\right. C^{c} \left.\right) = 0 . 2

Show whether or not the pair $\left\{A , B\right\}$ is independent.

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5.3 Exercise 2

Suppose $\left\{A_{1} , A_{2} , A_{3}\right\} ci \left|\right. C$ and $ci \left|\right. C^{c}$ , with $P \left(C\right) = 0 . 4$ , and

P \left(\right. A_{i} \left|\right. C \left.\right) = 0 . 90 , 0 . 85 , 0 . 80 P \left(\right. A_{i} \left|\right. C^{c} \left.\right) = 0 . 20 , 0 . 15 , 0 . 20 \text{for} i = 1 , 2 , 3 , \text{respectively}

Determine the posterior odds $P \left(\right. C \left|\right. A_{1} A_{2}^{c} A_{3} \left.\right) / P \left(\right. C^{c} \left|\right. A_{1} A_{2}^{c} A_{3} \left.\right)$ .

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5.3 Exercise 3

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:

Good weather (no…

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5.3 Exercise 5

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 rece…

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5.3 Exercise 6

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…

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5.3 Exercise 7

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

…

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5.3 Exercise 8

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

H be the event that a well would be successful
S be the event the geological conditions are favo…

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5.3 Exercise 9

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

H be the event the performance is satisfactory
S be th…

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5.3 Exercise 10

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.

$H =$ the event the program is satisfac…

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5.3 Exercise 11

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

H be the event the product is introduced and successful
S be the event the R&D group produces a product with the desired characteristics.
E be the event the testing program indicates the product is satisfactory

The company assumes $P \left(S\right) = 0 . 9$ and the conditional probabilities

P \left(\right. H \left|\right. S \left.\right) = 0 . 90 P \left(\right. H \left|\right. S^{c} \left.\right) = 0 . 10 P \left(\right. E \left|\right. S \left.\right) = 0 . 95 P \left(\right. E \left|\right. S^{c} \left.\right) = 0 . 15

Since the testing of the merchandise is not affected by…

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5.3 Exercise 12

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

$H =$ the event she will get the raise
$S =$ the event company profits increase by ten percent or more
$E =$ the event sales volume is up by fifteen percent or mo…

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5.3 Exercise 13

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 $A , B , C$ 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 $\left\{A , B , C\right\} ci \left|\right. H$ and

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5.3 Exercise 14

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…

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5.3 Exercise 15

In a region in the Gulf Coast area, oil deposits are highly likely to be associated with underground salt domes. If H is the event that an oil deposit is present in an area, and S is the event of a salt dome in the area, experience indicates $P \left(\right. S \left|\right. H \left.\right) = 0 . 9$ and $P \left(\right. S \left|\right. H^{c} \left.\right) = 0 . 1$ . Company executives believe the odds favoring oil in the area is at least 1 in 10. It decides to conduct two independent geophysical surveys for the p…

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5.3 Exercise 16

A sample of 150 subjects is taken from a population which has two subgroups. The subgroup membership of each subject in the sample is known. Each individual is asked a battery of ten questions designed to be independent, in the sense that the answer to any one is not affected by the answer to any other. The subjects answer independently. Data on the results are summarized in the following table:

…

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5.3 Exercise 17

The data of the exercise, above, are converted to conditional probabilities and probabilities, as follows (probabilities are rounded to two decimal places).

For the following profiles classify each individual in one of the subgroups.

y, n, y, n, y, u, n, u, y, u
n, n, u, n, y, y, u, n, n, y
y, y, n, y, u, u, n, n, y, y

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