Question

Use Bayes' theorem to solve this problem. A storeowner purchases stereos from two companies. From Company A, 550 stereos are purchased and 1% are found to be defective. From Company B, 850 stereos are purchased and 6% are found to be defective. Given that a stereo is defective, find the probability that it came from Company A.

Answer 1

Baye's theorem
P(P|Q)=P(P∩Q)/P(Q)=P(Q∩P)/P(Q)
=P(Q∩P)/P(P)*(P(P)/P(Q))
=P(Q|P)*(P(P)/P(Q))
or simply
P(P|Q)=P(Q|P)*(P(P)/P(Q))

Here
P=stereo came from company A
Q=stereo is defective

So
P(P|Q) probability that the stereo came from company A given that it is defective.

P(Q|P) probability that the stereo is defective given that it came from company A. (this value is given).

P(P)=probability that the stereo came from company A = 550/total no. bought
P(Q)=probability that a stereo is defective
=total number of defectives / total no. of stereos bought.

Plug in the above numbers to get the answer. Post for a check if you wish.

Answer 2

I plugged in the numbers and here's my answer...


Company A: 1% x 550 = 5.5 are bad

Company B: 6% x 850 = 51 are bad

Total bad of defective stereos = 5.5 + 51 = 56.5

P(A | Bad) = 5.5 / 56.5 = 11/113

So Probability that the defective one is from A = 0.097345