Jane and Bob are on the same game show. They answer one question, either True or False. The probability that Bob is correct is 70%. The probability that Jane is correct is also 70%. Moreover, the probability that Bob is correct given Jane is either correct or incorrect is still 60%. What is the probability that Bob and Jane are correct given they both give the same answer?
This is how I've done it P(B)=0.70, P(J)=0.70. Then from the third piece of info I have P(B given (J U J')) but that doesn't seem to make sense to me because Bob being correct isn't affected by Jane being correct or not? So is the third statement just P(B n J) + P(B n J') = 0.60 but then isn't that just P(B) which is not 0.60 but 0.70?
To solve this problem, let's break it down step by step:
1. Let's consider the probability that Bob is correct (P(B)) and Jane is correct (P(J)). Given that Bob's probability of being correct is 70%, we have P(B) = 0.70.
Given that Jane's probability of being correct is also 70%, we have P(J) = 0.70.
2. Now, let's interpret the third piece of information. It states that the probability that Bob is correct, given that Jane is correct or incorrect, is still 60%. In other words, we are given that P(B | J U J') = 0.60.
However, you correctly pointed out that Bob's correctness is irrespective of Jane's correctness. So, we can assume that P(B | J U J') should be the same as P(B). This implies that P(B) = 0.60.
3. But this contradicts the original information that P(B) = 0.70. So, there seems to be an inconsistency in the problem statement.
Based on the given information, it's not possible to determine the probability that both Bob and Jane are correct, given that they give the same answer. More information is needed to solve the problem accurately.
I recommend double-checking the problem statement or providing any additional information if available, so we can proceed with a correct calculation.