conditional probability

Conditional probability

$A,B$ are two events (outcomes) of a random experiment. The probability of an event $A$ given that another event $B$ has occurred is:

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

If we know that $B$ has occured, $B$ becomes the new $\Omega$:
→
In this case, $A$ becomes a lot more likely, as it occupies a larger fraction of the sample space than it did before.