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.
11/113
To solve this problem using Bayes' theorem, we need to assign some variables:
Let's define:
- A: The event that a stereo is purchased from Company A.
- B: The event that a stereo is purchased from Company B.
- D: The event that a stereo is defective.
We are trying to find the probability that a stereo came from Company A given that it is defective, P(A|D).
Bayes' theorem states that:
P(A|D) = (P(D|A) * P(A)) / P(D)
Let's calculate each of these probabilities step by step:
1. P(D|A): The probability that a stereo is defective given that it is purchased from Company A. We know that 1% of Company A's stereos are defective, so P(D|A) = 0.01.
2. P(A): The probability that a stereo is purchased from Company A. We are given that the storeowner purchases 550 stereos from Company A and 850 stereos from Company B. Therefore, P(A) = 550 / (550 + 850).
3. P(D): The probability that a stereo is defective. This can be calculated using the law of total probability. It can be expressed as:
P(D) = P(D|A) * P(A) + P(D|B) * P(B)
We know that 1% of Company A's stereos are defective and 6% of Company B's are. Let's calculate P(D|B):
P(D|B) = 0.06
Now, let's calculate P(B):
P(B) = 850 / (550 + 850)
Now that we have all the necessary values, we can calculate P(D):
P(D) = P(D|A) * P(A) + P(D|B) * P(B)
Finally, we can calculate P(A|D) using Bayes' theorem:
P(A|D) = (P(D|A) * P(A)) / P(D)
Plug in the values you calculated to get the final answer.