5.2 Problems on conditional independence  (Page 3/4)

A research group is contemplating purchase of a new software package to perform some specialized calculations. Thesystems 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 satisfactory for the intended application
• $S=$ the event the program is free of significant bugs
• $E_1=$ the event the first diagnostic tests are satisfactory
• $E_2=$ the event the second diagnostic tests are satisfactory

Since the tests are for the presence of bugs, and are operationally independent, it seems reasonable to assume $\{H,E_1,E_2\}\mathrm{ci}|S$ and $\{H,E_1,E_2\}\mathrm{ci}|S^c$ . Because of the reliability of the software company, the manager thinks $P\left(S\right)=0.85$ . 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

• 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

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

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'snet 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 more

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

End of the year records show that sales increased by eighteen percent. What isthe 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 $A,B,C$ be the events the respective consultants agree this is the case. The physician decides togo with the majority. Since the advisers act in an operationally independent manner, it seems reasonable to suppose $\{A,B,C\}\mathrm{ci}|H$ and $\mathrm{ci}|H^c$ . 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 isnot present)?