Conditional Probability

Conditional probability recalculates chance within the smaller sample space where given information is already known to be true.

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Events are subsets of a sample space, and conditional probability changes the reference set. A sample-space rectangle contains two overlapping event regions and a small two-stage probability tree. The overlap marks joint occurrence, while branch labels show how later probabilities can depend on an earlier outcome.

“Given that” changes the denominator: only outcomes consistent with the known condition remain possible.

Definition

For events $A$ and $B$ with $P(B)>0$,

P(A\mid B)=\frac{P(A\cap B)}{P(B)}.

Read $P(A\mid B)$ as “the probability of $A$ given $B$.” The condition $B$ becomes the new sample space.

Restricted-sample-space idea

Once $B$ is known to have occurred, outcomes outside $B$ are irrelevant. Among the outcomes in $B$, count or measure the portion also belonging to $A$.

This explains why $P(A\mid B)$ and $P(B\mid A)$ are usually different: they use different denominators.

Worked example from counts

The denominator is $36$, not the original $60$.

Multiplication rule

Rearrange the conditional formula:

P(A\cap B)=P(A\mid B)P(B).

Equivalently,

P(A\cap B)=P(B\mid A)P(A).

This rule is useful for sequential events and tree diagrams.

Bayes' rule

Bayes' rule reverses a condition:

P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}.

It combines a prior probability with evidence. In testing contexts, a high test accuracy does not by itself determine the chance a person has the condition; the base rate also matters.

Common mistakes

Using the whole sample space as denominator after “given.” Restrict to the condition.

Swapping $P(A\mid B)$ and $P(B\mid A)$. Their denominators differ.

Adding probabilities along a tree path. Multiply along; add separate paths.

Calling mutually exclusive events independent. Positive-probability exclusive events affect each other completely.

Assuming independence without a calculation or design reason. Test it.