To find the probability that he knew the answer to the question given that he answered it correctly, we can employ Bayes’ theorem. Bayesian probability allows us to update our beliefs based on new evidence. In this case, we’re interested in the conditional probability.
Let’s define the following events:
- K: He knew the answer.
- C: He answered correctly.
We are seeking P(K | C), which is the probability he knew the answer given that he answered correctly. According to Bayes’ theorem, this can be calculated using the formula:
P(K | C) = (P(C | K) * P(K)) / P(C)
Where:
- P(C | K) is the probability of answering correctly given that he knew the answer.
- P(K) is the prior probability that he knew the answer.
- P(C) is the total probability of answering correctly.
Let’s analyze each component:
- If he knows the answer, we can assume that he answers correctly (P(C | K) = 1).
- The prior probability P(K) might depend on the context, say let’s assume it’s 0.6 (60% chance he knows the answer).
- Now, the probability of answering correctly, P(C), can be calculated using the law of total probability:
P(C) = P(C | K) * P(K) + P(C | ¬K) * P(¬K)
Where:
- P(C | ¬K) is the probability of answering correctly given that he did not know the answer. Suppose it is 0.2 (20% chance of guessing correctly).
- P(¬K) is the probability that he did not know the answer, which would be 1 – P(K) = 0.4 (40% chance he does not know).
Now plugging in the values:
- P(C) = (1 * 0.6) + (0.2 * 0.4) = 0.6 + 0.08 = 0.68.
Now we can substitute these values back into Bayes’ theorem:
P(K | C) = (1 * 0.6) / 0.68
P(K | C) ≈ 0.8824
This means that given that he answered the question correctly, there is approximately an 88.24% chance that he knew the answer. This kind of analysis is particularly useful in various scenarios, from educational testing to decision-making processes, helping us understand the likelihoods based on available information.
In summary, leveraging Bayes’ theorem in this way can provide a clearer picture of the probabilities involved based on what we know, allowing us to make more informed conclusions.