To determine the probability that he knew the answer given that he answered it correctly, we can use Bayes’ theorem.
Let:
- K be the event that he knew the answer.
- C be the event that he answered correctly.
According to Bayes’ theorem:
P(K | C) = (P(C | K) * P(K)) / P(C)
Where:
- P(K | C) is the probability that he knew the answer given that he answered correctly.
- P(C | K) is the probability of answering correctly given that he knew the answer.
- P(K) is the probability that he knew the answer.
- P(C) is the total probability of answering correctly.
To apply this formula, we need to know or estimate these probabilities:
- If he is knowledgeable about the topic, P(C | K) is likely high, approaching 1.
- P(K) is the overall likelihood that he knew the answer beforehand, which could vary based on the context.
- P(C) can be derived from the overall chance of answering the question correctly, factoring in both those who knew the answer and those who did not.
Once we have all these probabilities, we can substitute them back into the equation to find P(K | C), which gives us the desired probability.