To solve this problem using Bayes' theorem, we need to calculate the probability that a stereo came from Company A given that it is defective.
Let's define the following events:
A = The stereo came from Company A
B = The stereo came from Company B
D = The stereo is defective
We are given the following probabilities:
P(A) = Probability of choosing a stereo from Company A = 550 / (550 + 850) = 0.393
P(B) = Probability of choosing a stereo from Company B = 850 / (550 + 850) = 0.607
P(D|A) = Probability that a stereo is defective given it came from Company A = 0.01
P(D|B) = Probability that a stereo is defective given it came from Company B = 0.06
We need to find P(A|D), which is the probability that a stereo came from Company A given that it is defective. Applying Bayes' theorem, we can write:
P(A|D) = (P(D|A) * P(A)) / P(D)
To calculate P(D), which is the probability of a stereo being defective, we can use the law of total probability:
P(D) = P(D|A) * P(A) + P(D|B) * P(B)
Substituting the given values:
P(D) = (0.01 * 0.393) + (0.06 * 0.607)
Simplifying, we find:
P(D) = 0.00393 + 0.03642
P(D) = 0.04035
Now, substituting the values back into Bayes' theorem:
P(A|D) = (0.01 * 0.393) / 0.04035
Simplifying further, we find:
P(A|D) = 0.00393 / 0.04035
P(A|D) ≈ 0.0974
Therefore, the probability that a defective stereo came from Company A is approximately 0.0974, or 9.74%.