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Back to 5.3 Problems on Conditional Independence

5.3 Exercise 9

Avg difficulty 2.5

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

  1. H be the event the performance is satisfactory
  2. S be the event the system satisfies customer acceptance tests
  3. E1 be the event the first benchmark tests are satisfactory.
  4. E2 be the event the second benchmark test is ok.

Under the usual conditions, we may assume {H,E1,E2} ci |S\left\{H , E_{1} , E_{2}\right\} ci \left|\right. S and ci |Scci \left|\right. S^{c}. Reliability data show

P ( H | S ) = 0 . 95 , P ( H | S c ) = 0 . 45 P \left(\right. H \left|\right. S \left.\right) = 0 . 95 , P \left(\right. H \left|\right. S^{c} \left.\right) = 0 . 45
P ( E 1 | S ) = 0 . 90 P ( E 1 | S c ) = 0 . 25 P ( E 2 | S ) = 0 . 95 P ( E 2 | S c ) = 0 . 20 P \left(\right. E_{1} \left|\right. S \left.\right) = 0 . 90 P \left(\right. E_{1} \left|\right. S^{c} \left.\right) = 0 . 25 P \left(\right. E_{2} \left|\right. S \left.\right) = 0 . 95 P \left(\right. E_{2} \left|\right. S^{c} \left.\right) = 0 . 20

Determine the posterior odds P(H|E1E2)/P(Hc|E1E2)P \left(\right. H \left|\right. E_{1} E_{2} \left.\right) / P \left(\right. H^{c} \left|\right. E_{1} E_{2} \left.\right).

Pfeiffer, Applied Probability, 5.3 Problems on Conditional Independence, exercise 9

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.