Bayes Rule Approach

Question

The city of San Fransokyo is a high-tech city, especially in the field of robotics. There are three big companies that rule the city: Granville, Axcel, and Hamada. The Granville company controls about 50% of the total robot manufacturing in the city of San Fransokyo; Axcel company controls 35%; while the rest is controlled by the Hamada company. Based on the information you have obtained, it turns out that 3% of the robots made by Granville have problems with the sensing ability of the robot. As much as 5% robots made by Axcel and 1% robots made by Hamada also have the same problem. You want to buy a robot to help with your business and randomly pick up exactly one robot from the three companies. After the item arrives, you do a test, and it turns out that the robot is experiencing a malfunction in its sensing component. Using the Bayes Rule approach, which company is predicted to make the robot you bought? Justify your answer with adequate calculations!

Answer 1

You are given P(G) = .5, P(A)=.35, and P(H) = .15

Let p represent problems. You are also given P(p|G) = .03, P(p|A) = .05, and P(p|H) = .01

You want to find P(Company|p) Let's look at G:

Bayes Law: P(G|p) = P(p|G)P(G)/P(p) We are given both terms in the numerator

P(p) = P(G)P(p|G) + P(A)P(p|A) + P(H)P(p|H) = .5(.03) + .35(.05) + .15(.01) = .034

P(G|p) = .5(.03)/.034 = .441

You can figure out P(A|p) and get the last one by subtracting the two probabilities from 1.