Question

16. Your large corporation manufactures a certain popular brand of robot lawn mower. After manufacture, all mowers go through a standard quality control check. The Portland factory, which manufactures 30% of your mowers, has a probability of .8 that each mower will pass the quality control check. The Dallas factory, which manufactures 50% of your mowers, has a probability of .7 that each mower will pass the quality control check. For the Buffalo factory,
which manufactures the remaining 20% of your mowers, you have been unable to learn what the probability is that each mower will pass inspection.   But, the corporation’s annual report claims that the overall probability that one of their manufactured mowers will pass the quality control check is .72  

a) What do you conclude is the probability that a mower manufactured at the Buffalo factory will pass the quality control check?  
b) A customer orders one of your mowers, and of course receives one that has passed the quality control check. What is the probability it was manufactured in Dallas?  
     

Answer 1

a)

Let P, D and B be the event that the mower is manufactured at Portland, Dallas and Buffalo factory respectively.

Let Q be the event that the mower has passed the quality control check.

Given,

Pr(P) = 0.3, Pr(D) = 0.5 and Pr(B) = 0.2

Pr(Q | P) = 0.8 Pr(Q | D) = 0.7   

Pr(Q) = 0.72

By law of total probability,

Pr(Q) = Pr(P) Pr(Q | P) + Pr(D) Pr(Q | D) + Pr(B) Pr(Q | B)

0.72 = 0.3 * 0.8 + 0.5 * 0.7 + 0.2 Pr(Q | B)

Pr(Q | B) = (0.72 - 0.59)/0.2

Probability that a mower manufactured at the Buffalo factory will pass the quality control check

= Pr(Q | B) = 0.65

(b)

Given that the mower has passes the quality control check, probability it was manufactured in Dallas

= Pr(D | Q)

= Pr(Q | D) Pr(D) / Pr(Q) (Bayes theorem)

= 0.7 * 0.5 / 0.72

= 0.4861