Probability- defective product and market share

There are three manufactures.

Company A with 70% market share
company B with 20% market share
Company C with 10% market share

2% of Company A's products are defective
5% of company B's products are defective
10% of Company C's products are defective

what is the probability of a defective product coming from company A?

I have no idea how to do this one. can some one explain it to me?

thanks

WlND said:

There are three manufactures.

Company A with 70% market share
company B with 20% market share
Company C with 10% market share

2% of Company A's products are defective
5% of company B's products are defective
10% of Company C's products are defective

what is the probability of a defective product coming from company A?

I have no idea how to do this one. can some one explain it to me?

thanks

Click to expand...

How do defective parts make it to market:
P(A) = 0.70,.....P(defective | A) = 0.02
P(B) = 0.30,.....P(defective | B) = 0.05
P(C) = 0.10,.....P(defective | C) = 0.10

P(defective) = P(defective | A)*P(A) + P(defective | B)*P(B) +P(defective | C)*P(C)
.................= P(A & defective) + P(B & defective) + P(C & defective)
.................= . . .

Then use Bayes's Theorem:

P(A | defective) = P(A & defective) / P(defective)
etc.

Imagine 100000 products. "Company A with 70% market share" so 70000 are from company A."2% of Company A's products are defective" so .02(70000)= 14000 of these are defective.

"Company B with 20% market share" so 20000 products are from company B. "5% of company B's products are defective" so .05(20000)= 1000 of these are defective.

"Company C with 10% market share" so 10000 products are from Company C. "10% of company C's products are defective" so .10(10000)= 1000 of these are defective.

Of a total of 14000+ 1000+ 1000= 16000 defective products, 14000 are from company A so, given that a product is defective, the probability it is from company A is \(\displaystyle \dfrac{14000}{16000}= 0.875\) or 87.5%.

Of a total of 14000+ 1000+ 1000= 16000 defective products, 1000 are from company B so, given that a product is defective, the probability it is from company B is \(\displaystyle \dfrac{1000}{16000}= 0.0625\) or 6.25%.

Of a total of 14000+ 1000+ 1000= 16000 defective products, 1000 are from company C so, given that a product is defective, the probability it is from company C is \(\displaystyle \dfrac{1000}{16000}= 0.0625\) or 6.25%.