A construction company employs two sales engineers. Engineer 1 does the work of estimating cost for 70% of jobs bid by the company. Engineer 2 does the work for 30% of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0.02 is the probability of an error when he does the work, whereas the probability of an error in the work of engineer 2 is 0.04. Suppose a bid arrives and a serious error occurs in estimating cost. Which engineer would you guess did the work? Explain and show all work.

Then P( E | 1 ) is the probability that an error occurs when engineer 1 does the work and P( E | 2 ) is the probability that an error occurs when engineer 2 does the work.

To have an idea of which engineer is more likely to do the work when an error occurs you need to calculate P( 1 | E ) and P( 2 | E ), The probability that engineer 1 does the work when an error occurs and the probability that engineer 2 does the work when an error occurs.

The Bayes's theorem states:

[tex]P( B | A ) = \frac{P( A | B )*P( A )}{P( B )}[/tex]

Using the notation above:

[tex]P( 1 | E ) = \frac{P( E | 1 )*P( 1 )}{P( E )}[/tex]

P( E ) = 0.7*0.02 + 0.3*0.04 = 0.026 /// the probability that engineer 1 does the work and an error occurs or the probability that engineer 2 does the work and an error ocurrs.

[tex]P( 1 | E ) = \frac{0.04*0.3}{0.026}=0.538[/tex]

Doing the same for engineer 2:

[tex]P( 2 | E ) = \frac{0.04*0.3}{0.026}=0.462[/tex]