Conditional Probability (PS1 – 5.3)

Core Concept

What "conditional" means

The notation $P(A \mid B)$ is read as "the probability of $A$ given $B$". It represents the probability that event $A$ occurs, under the condition that event $B$ is known (or assumed) to have occurred. Conditioning effectively restricts the sample space to only those outcomes in $B$, and asks what fraction of those outcomes also lie in $A$.

Intuition with a sample space: Suppose you roll two fair dice and record the outcomes as ordered pairs. There are 36 equiprobable elementary events. If you are told that the sum is at least 10, the sample space shrinks to $\{(4,6),(5,5),(5,6),(6,4),(6,5),(6,6)\}$ — just 6 outcomes. The conditional probability of "both dice show the same number given sum $\geq 10$" is $\frac{1}{6}$ (only $(5,5)$ qualifies out of those 6).

The formula

This formula is always valid. It is especially useful when you cannot list all elementary events but you do know (or can calculate) $P(A \cap B)$ and $P(B)$.

Rearranging gives the Multiplication Rule (a prerequisite topic):

which is exactly what each branch-pair on a tree diagram computes.

Independence as a special case

Events $A$ and $B$ are independent if and only if $P(A \mid B) = P(A)$. Knowing $B$ has occurred gives no information about $A$. This is consistent with the multiplication rule for independent events: $P(A \cap B) = P(A) \times P(B)$.