In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% respectively of the total output of bolts. Of their outputs, 5%, 4% and 2% respectively are defective. A bolt is chosen at random from the factory's output and found to be defective. What is the probability that it came from machine B?
Question
In a bolt factory, machines A, B and C manufacture 25%, 35% and 40% respectively of the total output of bolts. Of their outputs, 5%, 4% and 2% respectively are defective. A bolt is chosen at random from the factory's output and found to be defective. What is the probability that it came from machine B?
Solution
To solve this problem, we can use Bayes' theorem, which is a way of finding a probability when we know certain other probabilities. The formula is:
P(A|B) = [P(B|A) * P(A)] / P(B)
Where:
- P(A|B) is the probability we are trying to find: the bolt is from machine B given that it is defective.
- P(B|A) is the probability that a bolt is defective given that it is from machine B.
- P(A) is the probability that a bolt is from machine B.
- P(B) is the total probability that a bolt is defective.
Let's plug in the values:
- P(A) = 35% = 0.35 (the probability that a bolt is from machine B)
- P(B|A) = 4% = 0.04 (the probability that a bolt is defective given that it is from machine B)
- P(B) is the total probability that a bolt is defective, which can be calculated as follows:
- P(B) = P(B and A) + P(B and not A)
- P(B and A) = P(B|A) * P(A) = 0.04 * 0.35 = 0.014
- P(B and not A) = P(B and machine A) + P(B and machine C) = (0.05 * 0.25) + (0.02 * 0.40) = 0.0125 + 0.008 = 0.0205
- So, P(B) = 0.014 + 0.0205 = 0.0345
Now we can plug these values into Bayes' theorem:
P(A|B) = (0.04 * 0.35) / 0.0345 = 0.014 / 0.0345 = 0.4058
So, the probability that a randomly chosen defective bolt came from machine B is approximately 0.406 or 40.6%.
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