How to Calculate Conditional Probability

Calculates the probability that A happens given that B happened, P(A|B) = P(A∩B) ÷ P(B). It is the share of A within the world where B has occurred.

Explanation

The conditional probability $P(A \mid B)$ is the chance that $A$ happens given that $B$ has happened.

P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}

Here $P(A \cap B)$ is the probability that $A$ and $B$ both occur.

The idea is to shrink the world down to $B$ . Treat the outcomes where $B$ happened as the new whole, and measure what fraction of them also contain $A$ .

Example

If $P(A \cap B) = 0.2$ and $P(B) = 0.5$ :

P(A \mid B) = \dfrac{0.2}{0.5} = 0.4

Restricted to the cases where $B$ occurred, $A$ happens 40% of the time. Whatever the unconditional probability of $A$ was, learning that $B$ happened has updated it.

Independence

When $P(A \mid B) = P(A)$ , the events $A$ and $B$ are independent: knowing about $B$ tells you nothing about $A$ .

In that case $P(A \cap B) = P(A)P(B)$ , so the probabilities simply multiply. The converse is the trap: multiplying probabilities that are not independent gives the wrong answer.

Watch out

The formula breaks down when $P(B) = 0$ ; you cannot condition on something that never happens
$P(A \mid B)$ and $P(B \mid A)$ are completely different quantities. "The chance that a sick person tests positive" and "the chance that someone testing positive is sick" are famously far apart