conditional probability with false positives

Question

I have a homework question and I'm not sure how to approach it. Straight formula or tree?

To me, 0.05 the false negative. But 0.96 does not seem like a false positive.

Question:
Your company is producing steel beams and the probability that a beam will be faulty is 0.08. The company is also experimenting with a new artificial intelligence (AI) test to examine whether a beam is faulty or not. The probability that the test will be accurate when the beam is indeed faulty is 0.96 but experience has shown that there is a chance of 0.05 for the AI test to wrongly classify a beam as faulty when it is not.

1. Calculate the probability that a beam will be correctly classified by the AI test as not faulty

2. Calculate the probability that a beam, following the AI test will be classified as faulty

3. Given that on a randomly chosen day the company is producing and testing 1,000 beams, how many of them will be classified by the AI test as faulty and will indeed be faulty?

Answer 1

You're given three probabilities:

• $P(F)=0.08$
• $P(\text{AI is correct}|F)=0.96$, this is True Positive.
• $P(\text{Ai is wrong}|F')=0.05$, this is False Negative because the beam is indeed not fault but AI says it is.

For each of the question, you need to formulate what it's being asked, and try to find them in terms of above listed probabilities:

1. $P(\text{AI is correct and beam is not Faulty})=P(\text{AI is correct}|F')P(F')=(1-P(\text{AI is wrong}|F'))P(F')$