My textbook includes this question:
If $A$ and $B$ are events such that $P(A|B) = P(B|A)$, then
$(A) A \subset B$ but $A \neq B$
$(B) A = B$
$(C) A \cap B = \emptyset$
$(D) P(A) = P(B)$
The answer key states only option $(D)$ as the answer. Is option $(C)$ also correct since if $A \cap B = \emptyset$, then $P(A \cap B) = P(\emptyset) = 0$ and putting that into the conditional probability formula makes both sides $P(A|B) = P(B|A) = 0$? Please comment. Thank you
I believe the book is incorrect and there is not enough information to answer this question adequately.
We are told $A$ and $B$ are events such that $P(A|B) = P(B|A)$, and this is the only assumption we are allowed to make. Under this assumption, we know
$$\frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)}$$
by definition of conditional probability. This statement holds whether $A$ and $B$ are disjoint, so we do not know whether $A$ and $B$ are disjoint. In other words, we do not know if $A \cap B = \emptyset$.
If $A$ and $B$ are not disjoint with $P(A \cap B) > 0$, then division by $P(A \cap B)$ is defined and we can derive $P(A) = P(B)$ as follows
$ \frac{1}{P(A \cap B)} \cdot \frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)} \cdot \frac{1}{P(A \cap B)}$
$\Leftrightarrow \frac{1}{P(B)} = \frac{1}{P(A)}$
$\Leftrightarrow P(A)P(B) \cdot \frac{1}{P(B)} = \frac{1}{P(A)} \cdot P(A)P(B)$
$\Leftrightarrow P(A) = P(B)$
But if $A$ and $B$ are disjoint with $P(A \cap B) = P(\emptyset) = 0$, then division by $P(A \cap B)$ is not defined and $P(A) = P(B)$ is not necessarily true. For instance, imagine we flip a biased coin with events $H$ and $T$ for heads and tails, respectively. Of course, $H \cap T = \emptyset$, so we have $P(H|T) = P(T|H) = 0$, and since the coin is biased, we have $P(H) \neq P(T)$. However, if the coin is fair, then $P(H|T) = P(T|H) = 0$ and $P(H) = P(T)$.
Hence, $(C)$ cannot be an answer because we are not told whether $A$ and $B$ are disjoint and this is not implied by the information given. And $(D)$ cannot be an answer because it depends on whether $A$ and $B$ are disjoint.