SL 4.11—Conditional and independent probabilities, test for independence

In everyday life, new information changes what we expect. If you know it's already raining, the probability that someone will carry an umbrella is different from the baseline probability. This is the essence of conditional probability.

Conditional Probability: The probability of event $A$ occurring given that event $B$ has already occurred. It is written as $P(A|B)$, read as "the probability of $A$ given $B$".

The formula is:

$P(A|B)=\frac{P(A\cap B)}{P(B)},P(B)\ne 0$

Where:

• $P(A\cap B)$ is the probability that both $A$ and $B$ occur
• $P(B)$ is the probability that the conditioning event $B$ occurs

Intuitively, conditioning on $B$ restricts your sample space to only the outcomes where $B$ has happened , then you ask what fraction of those outcomes also include $A$.