Question

An anti-cheat program reveals that a text is AI-generated with a probability of 0.92. If the text is not AI-generated, the program will err and determine that the text is AI-generated with probability 0.07. It is reasonable to assume that the probability that a random student uses

AI to write a text is 0.05.

a) If the program concludes that the student's text is AI-generated, what is the probability that it actually is?

b) How small must the probability of the program wrongly determining that the text is AI-generated be, for the probability in a) to be over 90%?

Answer 1

a) The probability that the student's text is actually AI-generated given that the program concludes it is can be calculated using Bayes' theorem. b) To make the probability in part a) over 90%, we need to find the minimum probability of the program wrongly determining the text as AI-generated.

Step-by-step explanation:

a) To calculate the probability that the student's text is AI-generated given that the program concludes it is, we can use Bayes' theorem. Let A represent the event of the text being AI-generated and B represent the event of the program concluding it is AI-generated. According to Bayes' theorem, the probability that the text is AI-generated given that the program concludes it is, P(A|B), can be calculated as: P(A|B) = (P(B|A) * P(A)) / P(B). From the information given, P(B|A) = 0.92, P(A) = 0.05, and P(B) = 0.92 * 0.05 + 0.07 * (1 - 0.05). Plugging in these values will give us the desired probability.

b) To find the minimum probability of the program wrongly determining the text as AI-generated for the probability in part a) to be over 90%, we need to consider the complement probability. Let C represent the event of the program concluding the text is not AI-generated. The complement of the event A is A'. According to Bayes' theorem, the probability that the text is AI-generated given that the program concludes it is, P(A|B), can be calculated as: P(A|B) = (P(B|A) * P(A)) / P(B). From the information given, P(B|A) = 0.92, P(A) = 0.05, and P(B) = 0.92 * 0.05 + 0.07 * (1 - 0.05). We need to find the value of P(C) such that P(A|B) > 0.90.