A particular server may be working or not working, and you may assume that it is

Question

A particular server may be working or not working, and you may assume that it is either one or the other throughout the problem. If the server is not working, any attempt to access it fails. Even if the server is working, an attempt to access it can fail due to network congestion beyond the control of the server. Suppose that the a priori probability that the server is working is 3/4. Suppose that if the server is working, then each access attempt is successful with probability 2/3, independently of other access attempts. Find the probability that the second access attempt fails, given that the first access attempt fails. attempts fail. Find the probability that the server is working, given that the first and second access attempt fail.

Explanation / Answer

(a) Let A denote the event that the server is not working and B is the probability that the server is working.

Note that A and B are mutually exclusive events i.e. they cannot take place togerther.

Now, in order to find out the probability that the 2nd attempt fails given the first attempt also fails we need to cosider to facts seperately: 1. Server is not working and denote the corresponding total probability as P(A) and 2. Server is working and denote the corresponding probability as P(B).

P(A)=(1-3/4)(1-3/4)=1/16

P(B)=(3/4)(1-2/3)(1-2/3)=(3/4)(1/3)(1/3)=1/12

So, total probability P=P(A)+P(B)=1/16+1/12=7/48

(b) Server is working and first and 2nd attempt fails: this event can be reduced by the following chain of events: Server working and the first attempt fail->2nd attempt is made and it also fails.

P=(Prob. of server is working)(Prob that first attempt fails)=(3/4)(1-2/3)=1/4

Ptotal=PX(Prob that 2nd attempt fails)=(1/4)(1-2/3)=1/12

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