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Conditional probability
The conditional probability of $B$ given $A$ is
$$P(B\mid A)=\frac{P(A\cap B)}{P(A)}, \quad P(A)>0.$$
Independent events
Events $A$ and $B$ are independent if knowing that $A$ happened does not change the probability of $B$. Formally, $P(B\mid A)=P(B)$.
Mutually exclusive
Two events are mutually exclusive if they cannot happen at the same time. In set notation, $A\cap B=\varnothing$, so $\mathrm{P}(A\cap B)=0$.