What is the probability that a serious error was made by Engineer 1 or 3?

[n77ler]

Homework Statement

A construction company employs three sales engineers. Engineers 1,2 nd 3 estimate the costs of 30%, 20%, and 50%, respectively, of all jobs bid by the company. For i=1,2,3, define E_i to be the event that a job is estimated by engineer i. The following probabilities describe the rates at which the engineers make serious errors in estimating costs:

P(error/E_1)= 0.01, P(error/E_2)= 0.03, P(error/E_3)

a.) If a particular bid results in a serious erros in estimating job cost, what is the probabilty that the error was made by engineer 1?
b.)If a particular bid results in a serious error in estimating job cost, what is the probabilty that the error was made by engineer 3?

Homework Equations

I drew a tree diagram with 3 branches the first three branches were E1, E2, E3 and they were 0.3, 0.2, and 0.5 for the costs that they had to estimate. It then branches for each one of them and shows the probability of the serious errors in estimating... E1, E2, and E3 were 0.01, 0.03 and 0.02 respectively.

The Attempt at a Solution

a.) I used P(E_1/Error) so given that serious error occurs what is the probability of it being Engineer 1.

=[ P(E_1 union Error) ] / P(error)

 

[danago]

That looks alright so far. Now how can you find $$P(e \cap E_1 )$$ and $$P(e)$$? You can use your tree diagram, or perhaps take a more formal approach by looking into bayes theorem

 

[n77ler]

Ok I did it but forgot to finish it off on my post... It's =[ (0.3)x(0.01) ] / 0.46
P(Error) was worked out by multiplying 0.3, 0.2, 0.5 by 0.01, 0.03 ad the third error value
And then b will be the same thing!