The Breaker's Manufacturing Company makes 80% of a particular sensor, the Cartin Company makes 15% of them, and the Flutes Company makes the other 5%. The sensors made by Breaker's have a 4% defect rate, the Cartin sensors have a 6% defect rate, and the Flutes sensors have a 9% defect rate. If a sensor is randomly selected from the general population of all sensors, what is the probability that it is defective, given that it was made by Cartin Company?

Question

ratnakermehta20 · Answer

In the usual notation, we are given that :
P(B)=80/100, P(C)=15/100, P(F)=5/100.
P(D/B)=4/100, P(D/C)=6/100, P(D/F)=9/100.
So, if P(D/C) is the required prob. then, P(D/C)=6/100.

However, in case the required prob. is P(C/D) [ i.e., the probability that it is made by Cartin Company, given that it was defective?] then we proceed as under :

We have, P(D)=P(B)P(D/B)+P(C)P(D/C)+P(F)P(D/F),
=(80/100)(4/100)+(15/100)(6/100)+(5/100)(9/100),
i.e., P(D)=455/10000.

Hence, the reqd. prob.=P(C/D)={P(D/C)*P(C)}/P(D)={16/100*6/100}/(455/10000),
or, P(C/D)=90/455=18/91~~19.78%