TITLE: Conditional joint probability and independence
QUESTION [1 upvotes]: Let's have a joint probability of three events, $\mathbf{P}(X,A,B)$. If $\mathbf{P}(X|A) = \mathbf{P}(X)$, can we show that $\mathbf{P}(X|A,B) = \mathbf{P}(X|B)$? If so, how?

REPLY [3 votes]: Toss a fair coin twice, and let $A$ be the event that the first toss is a head and $B$ the event that the second toss is a head.
Now let $X$ be the event that the result of the two tosses is the same.
$P(X|A) = P(X|B)=P(X)=\dfrac 12$ but $P(X|A,B)=1$.
{"set_name": "stack_exchange", "score": 1, "question_id": 397204}