This is a classic problem of conditional probability. We are asked to find the probability that the student knew the answer given that he answered correctly. We can denote this as P(Knew | Correct).

We know the following probabilities:

P(Knew) = 0.8 (probability student knows the answer)
P(Guess) = 0.2 (probability student guesses)
P(Correct | Guess) = 0.25 (probability student is correct given that he guessed)

We also know that if the student knows the answer, he will definitely answer correctly. So, P(Correct | Knew) = 1.

We can use Bayes' theorem to solve this problem. Bayes' theorem states:

P(A | B) = P(B | A) * P(A) / P(B)

In this case, we want to find P(Knew | Correct), so we can substitute into Bayes' theorem:

P(Knew | Correct) = P(Correct | Knew) * P(Knew) / P(Correct)

We know everything on the right side except for P(Correct). We can find P(Correct) using the law of total probability:

P(Correct) = P(Correct and Knew) + P(Correct and Guess)
= P(Correct | Knew)P(Knew) + P(Correct | Guess)P(Guess)
= 1 * 0.8 + 0.25 * 0.2
= 0.8 + 0.05
= 0.85

Now we can substitute back into Bayes' theorem:

P(Knew | Correct) = 1 * 0.8 / 0.85
= 0.941

So, if the student correctly answers a question, the probability that he really knew the correct answer is approximately 0.941 or 94.1%.

Question

This is a classic problem of conditional probability. We are asked to find the probability that the student knew the answer given that he answered correctly. We can denote this as P(Knew | Correct).

We know the following probabilities:

P(Knew) = 0.8 (probability student knows the answer)
P(Guess) = 0.2 (probability student guesses)
P(Correct | Guess) = 0.25 (probability student is correct given that he guessed)

We also know that if the student knows the answer, he will definitely answer correctly. So, P(Correct | Knew) = 1.

We can use Bayes' theorem to solve this problem. Bayes' theorem states:

P(A | B) = P(B | A) * P(A) / P(B)

In this case, we want to find P(Knew | Correct), so we can substitute into Bayes' theorem:

P(Knew | Correct) = P(Correct | Knew) * P(Knew) / P(Correct)

We know everything on the right side except for P(Correct). We can find P(Correct) using the law of total probability:

P(Correct) = P(Correct and Knew) + P(Correct and Guess)
= P(Correct | Knew)P(Knew) + P(Correct | Guess)P(Guess)
= 1 * 0.8 + 0.25 * 0.2
= 0.8 + 0.05
= 0.85

Now we can substitute back into Bayes' theorem:

P(Knew | Correct) = 1 * 0.8 / 0.85
= 0.941

So, if the student correctly answers a question, the probability that he really knew the correct answer is approximately 0.941 or 94.1%.

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