Use Bayes' theorem to solve this problem. A storeowner purchases stereos from two companies. From Company A, 250 stereos are purchased and 1% are found to be defective. From Company B, 950 stereos are purchased and 10% are found to be defective. Given that a stereo is defective, find the probability that it came from Company A.

Question

Use Bayes' theorem to solve this problem. A storeowner purchases stereos from two companies. From Company A, 250 stereos are purchased and 1% are found to be defective. From Company B, 950 stereos are purchased and 10% are found to be defective. Given that a stereo is defective, find the probability that it came from Company A.

Answer 1

number of defective stereos = .01(250) + .1(950

= 2.5 + 95
= 97.5

so prob that the defective one is from A
= 2.5/97.5 = .02564

Answer 2

I did not know about Bayes Theorem, and just used common sense in my solution.

I looked it up and found this rather confusing description of it

http://en.wikipedia.org/wiki/Bayes%27_theorem

I don't think my old brain feels up to learning more about it, lol

Answer 3

To solve this problem using Bayes' theorem, we need to calculate the probability that a defective stereo came from Company A.

Let's define the following:

- Event A: A stereo is from Company A
- Event B: A stereo is defective

We need to find P(A|B), which represents the probability that a stereo is from Company A given that it is defective.

According to Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)

Now let's calculate each probability:

P(B|A): The probability that a stereo is defective, given that it is from Company A. In this case, we know that 1% of the stereos from Company A are defective, so P(B|A) = 0.01.

P(A): The probability that a randomly selected stereo is from Company A. Since we know that there are 250 stereos purchased from Company A out of a total of 250+950 = 1200 stereos purchased, we have P(A) = 250/1200 = 0.2083.

P(B): The probability that a randomly selected stereo is defective. To calculate this, we need to consider the sum of the defective stereos from both companies. Company A has 1% defective stereos out of 250 total stereos, so there are 0.01 * 250 = 2.5 defective stereos from Company A. Company B has 10% defective stereos out of 950 total stereos, so there are 0.1 * 950 = 95 defective stereos from Company B. The total number of defective stereos is 2.5 + 95 = 97.5. Therefore, P(B) = 97.5/1200 = 0.0813.

Using Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)
= (0.01 * 0.2083) / 0.0813
= 0.002083 / 0.0813
≈ 0.0256

Therefore, the probability that a defective stereo came from Company A is approximately 0.0256 or 2.56%.

Answer 4

To solve this problem using Bayes' theorem, we need to find the probability that a stereo is from Company A given that it is defective. Let's define the following events:

A: A stereo is from Company A
B: A stereo is from Company B
D: A stereo is defective

We are given the following information:

P(A) = probability that a stereo is from Company A = 250/1200 = 1/4
P(B) = probability that a stereo is from Company B = 950/1200 = 19/24
P(D|A) = probability that a stereo is defective given it is from Company A = 1%
P(D|B) = probability that a stereo is defective given it is from Company B = 10%

We want to find P(A|D), the probability that a stereo is from Company A given that it is defective.

By Bayes' theorem, we have:

P(A|D) = (P(D|A) * P(A)) / P(D)

To find P(D), the probability that a stereo is defective, we can use the law of total probability:

P(D) = P(D|A) * P(A) + P(D|B) * P(B)

Substituting the values, we have:

P(D) = (1/100) * (1/4) + (10/100) * (19/24)
= 1/400 + 19/240
= (6 + 19) / 240
= 25 / 240
= 5 / 48

Now, we can substitute this value into Bayes' theorem:

P(A|D) = (1/100) * (1/4) / (5/48)
= (1/400) / (5/48)
= (1/400) * (48/5)
= 12/2000
= 3/500

Therefore, the probability that a defective stereo came from Company A is 3/500.