In a computer lab 30 of the microcomputers are IBM 50 are Ap

In a computer lab, 30% of the microcomputers are IBM, 50% are Apple, and the rest are Dell microcomputers. Suppose that
at any given time the probability that a microcomputer works properly is 92% if it is IBM; 72% if it is Apple and 98% if it is
a Dell. One computer is selected at random from this lab. Let W be the event that selected computer is working. Find the
conditional probability that the selected computer is a Dell computer given that it is working properly. That is find P(Dell |W

Solution

As the probabilities of the computer brands add up to 100%, then P(Dell) = 0.20.

Let

I = IBM
A = Apple
D = Dell
W = working

Thus,

P(D|W) = P(D n W) / P(W)

As

P(D n W) = P(D) P(W|D) = 0.20*0.98 = 0.196

and

P(W) = P(I) P(W|I) + P(A) P(W|A) + P(D) P(W|D)

= 0.30*0.92 + 0.50*0.72 + 0.20*0.98

P(W) = 0.832

Thus,

P(D|W) = P(D n W) / P(W) = 0.235576923 [answer]