In a certain Engineering firm four engineers complete struct

In a certain Engineering firm, four engineers complete structural design projects. Steve works on 22% of the design projects, Nicole works on 46% of the design projects, Michelle works on 17% of the design projects, and Jose works on 15% of the design projects. The engineers are required to sign each of the design projects they work on. Review of the work reveals that projects with calculation errors for each of the four engineers are 2%, 3%, 4% and 1% respectively.

If a project was randomly chosen and found to contain calculation errors, Using Bayes\' Rule, what is the probability that this project was completed by Michelle?

Solution

Let

S, N, M J = the persons
E = error

hence

P(E) = P(S) P(E|S) + P(N) P(E|N) + P(M) P(E|M) + P(J) P(E|J)

P(E) = 0.22*0.02 + 0.46*0.03 + 0.17*0.04 + 0.15*0.01 = 0.0265

Thus,

P(M|E) = P(M) P(E|M) / P(E) = 0.17*0.04/0.0265 = 0.256603774 [ANSWER]