Finite Probability

Question

A computer chip manufacturer makes 60% of its computer chips in Arizona, 30% of its computer chips in California, and the remaining 10% of its computer chips in New Mexico. 2% of the computer chips made in Arizona are defective, 5% of the computer chips made in California are defective, and 7% of the computer chips made in New Mexico are defective. A recently purchased computer chip was defective. Find the probability that it was made in California. Round your answer to three decimal places.

Answer 1

Cassie,

 

       This problem is best done by starting with a tree with 3 branches, A,C &NM named for the states of manufacture. From each of the branches draw 2 branches representing D and notD. Put the fractions on each branch. So for example P(A)P(A/D)=.6*.02=.012 is the probability that a chip made in Arizona is defective. Do this for all 6 branches.

Now we want to calculate P(D/C), the probability that a defective chip was made in the California plant.

 

P(D/C)=P(C/D)*P(C)/P(D)  this is Bayes rule or theorem. You can calculate all the terms on the right hand side from the tree.

 

P(C/D)*P(D)=.3*.05=.015 and P(D)=P(A/D)+P(C/D)+P(NM/D)=.012+.015+.007=.034

 

So P(D/C)=.015/.034= .441

 

You can calculate the probabilities that a defective chip was made in either of the other two plants in a similar manner.

Hope this is not too confusing, it's simpler if you can see the tree.

 

Jim