# conditional probability with false positives

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?

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. $$P(\text{AI is correct and beam is not Faulty})=P(\text{AI is correct}|F')P(F')=(1-P(\text{AI is wrong}|F'))P(F')$$
• Ok thanks. But as you say, can you please show me how to find them. Thank you Aug 11, 2020 at 0:24
• I’ve shown the first one. Try the others and post it here if you’re stuck. Aug 11, 2020 at 5:44
• Sorry, but this is very advanced for me. If #1 is only asking Calculate the probability that a beam will be correctly classified by the AI test as not faulty, why would the formula be (1−𝑃(AI is wrong|𝐹′))𝑃(𝐹′)? Does that mean (1 - 0.05 * 0.92)=0.954? Is F' the complement of 0.08 which is 0.92? Sorry, but I'm very new to analytics and probability. I'm learning the best I can via Youtube and MOOCS. Aug 11, 2020 at 11:58
• Because "the beam is correctly classified by AI as not faulty" means the beam was not faulty AND AI is correct: P(AI is correct and beam is not faulty) = P(AI is correct | Not faulty)P(Not Faulty), as described above. P(F') is 0.92, 𝑃(AI is wrong|𝐹′) is 0.05, so the answer is (1-0.05) * 0.92 = 0.95 * 0.92 (beware of the parentheses) Aug 11, 2020 at 11:59
• So I was right! F' is the complement which is 0.92! Sweet Jesus. So the answer to #1 is 0.874. But to be honest I'm super confused with wording when it comes to probability questions. They always seem like riddles to me. Does that mean I'm not cut out for statistics? Maybe it's the fault of the university instructor? He gave us this problem, but he didn't even discuss the complement notation. Anyway, for #2 is the formula 𝑃(AI is correct and beam is faulty) = 𝑃(AI is correct|𝐹′)P(F) = (1 - P(AI is correct|F))P(F) -> (1 - 0.96) * 0.08 = 0.04 * 0.08 = 0.0032 = .32%? Aug 11, 2020 at 13:06