A webserver is in one of two states: working (W) or not working (W\'). The a pri

Question

A webserver is in one of two states: working (W) or not working (W'). The a priori probability that the webserver works is 0.8. Any attempt to access it is either successful (A) or a failure (A'). Assume that the state of the webserver remains fixed during any attempt to access it. If the webserver is working, any attempt to access it may fail due to network congestion, so in this case each access attempt is successful with probability 0.9. If the webserver is not working, any attempt to access it fails. If any attempt fails, another attempt to access the webserver is made. Find the following probabilities: (3) a). Find P(first access attempt fails). (4) b). Find P(server is working irst access attempt fails). (4) c). Find P(second access attempt fails first access attempt fails). (4) d). Find P(server is working first and second access attempts fail).

Explanation / Answer

P(W) = 0.8

P(W') = 0.2

P(A | W) = 0.9

P(A | W') = 0

P(A' | W) = 0.1

P(A' | W') = 1

a) P(A') = P(A' | W) * P(W) + P(A' | W') * P(W')

             = 0.1 * 0.8 + 1 * 0.2

             = 0.28

b) P(W | A') = P(A' | W) * P(W) / P(A') = 0.1 * 0.8 / 0.28 = 0.2857

c) P(second attemp fails | first attempt fails) = P(second attemp fails) * P(first attemp fails) / P(first attemp fails)

                                                                     = 0.28 * 0.28 / 0.28

                                                                     = 0.28

d) P(W | second attemp fails and first attemp fails) = 0.28572 = 0.0816

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