Refer to Part B of Table 2.3 for the conditional distribution of the number of network failures \(M\) given network age \(A\). Let \(\operatorname{Pr}(A=0)=0.5\); that is, you work in your room \(50 \%\) of the time.

a. Compute the probability of three network failures, \(\operatorname{Pr}(M=3)\).

b. Use Bayes' rule to compute \(\operatorname{Pr}(A=0 \mid M=3)\).

c. Now suppose you work in your room one-fourth of the time, so \(\operatorname{Pr}(A=0)=0.25\). Use Bayes' rule to compute \(\operatorname{Pr}(A=0 \mid M=3)\).

Table 2.3

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TABLE 2.3 Joint and Conditional Distributions of Number of Wireless Connection Failures (M) and Network Age (A) A. Joint Distribution M=0 M = 1 M = 2 M = 3 M = 4 Total Old network (A = 0) 0.35 0.065 0.05 0.025 0.01 0.50 New network (A = 1) 0.45 0.035 0.01 0.005 0.00 0.50 Total 0.80 0.10 0.06 0.03 0.01 1.00 B. Conditional Distributions of M given A Pr(M|A = 0) Pr(M|A = 1) M = 0 M=1 M = 2 M = 3 M=4 Total 0.70 0.13 0.10 0.05 0.02 1.00 0.90 0.07 0.02 0.01 0.00 1.00