$A$ given $B$ mutually exclusive with $D$?

Question

We are rolling a fair dice. Let A be the event rolling {1,2,3} and B the event{ 1,4} and D rolling {2,6}. Then can we say that (A given D) and the event B are mutually exclusive? Professor said that they are not but I think in order to events be mutually exclusive their intersection must be 0. And here if the event B occurred then it means we either rolled 1 or 4 and A given D means we either rolled 2 or 6 then clearly we can see they don’t have intersection.

Answer 1

"$A$ given $D$", i.e. $A \mid D$, is not an event; it is is a conditional event given $D$.

The event "we rolled $2$" is the event "$A$ and $D$", i.e. $A \cap D = \{2\}$, and that is indeed mutually exclusive with $B= \{1,4\}$, since $B$ and $D=\{2,6\}$ ("we rolled $2$ or $6$") are mutually exclusive