A computer has printer (P), disk (D), terminal (T), and magnetic tape (M) output

Question

A computer has printer (P), disk (D), terminal (T), and magnetic tape (M) outputs. Fifty-five percent of all output
characters are on D, twenty-five percent are on P, fifteen percent are on T, and five percent on M. The error rate for D is 1/2000, for P it is 2/1000, for T it is 1/1000, and for M it is 1/500. The experiment, E, is that a character is output and we observe (i) which type of device made that output and (ii) whether the character was correct. Write down the sample space. What is the probability the character was written on the disk, given A = {it was incorrect}? [Bayes rule].                                                                                                      This is a long question please read it carefully.

Explanation / Answer

Sample space is consists of the following 8 outcomes. Let C mean correct output and NC mean not correct output.

Then the sample space of E is { (P,C), (P,NC), (D,C), (D,NC), (T,C), (T,NC), (M,C), (M,NC) }.

We are given the following probabilities.

Pr(P) = .25

Pr(D) = .55

Pr(T) = .15

Pr(M) = .05

Pr(NC|P) = 2/1000

Pr(NC|D) = 1/2000

Pr(NC|T) = 1/1000

Pr(NC|M) = 1/500

Now we have to find Pr(D|NC).

By Bayes' theorem,

Pr(D|NC) = (Pr(NC|D) * Pr(D))/(Pr(NC|D)*Pr(D) + Pr(NC|P)*Pr(P) + Pr(NC|T)*Pr(T) + Pr(NC|M)*Pr(M))

= (0.000275)/(0.001025)

=0.26829 (repeating decimal)

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