The problem: Considering a couple, the probability of the man watching a specific TV program is 0.4, the probability of the woman watching the same program is 0.5, and the probability of the man watching any program that the woman is watching is 0.7.
Now what is the probability of the woman watching any program that the man is watching?
My textbook appears to have using the Bayes' formula in mind (the solution provided below), but to me this problem seems impossible to answer. Using the Bayes' formula we could get the probability of the woman watching the exact program mentioned in the question, when the man is watching it. And we can't just generalize it to any program as two of our probabilities are restricted to the specific program.
Is my thinking correct?
The provided solution: Having $M$ as the event of the man watching the mentioned program and $W$ as the event of the woman watching the mentioned program:
$P(W|M)=\frac{P(W)P(M|W)}{P(M)}=\frac{7}{8}$