Conditional Probability

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

Definition

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

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 $60$.

Two-way tables

A contingency table organizes intersections and marginal totals. For $P(A\mid B)$:

1. locate the row or column total for $B$;
2. locate the cell where $A$ and $B$ overlap;
3. divide the intersection count by the condition total.

Label categories carefully so complements and totals remain consistent.

Multiplication rule

Rearrange the conditional formula:

Equivalently,

This rule is useful for sequential events and tree diagrams.

Tree diagrams

Branches show conditional probabilities after earlier outcomes. Multiply along a path to find an intersection probability, and add disjoint path probabilities to combine outcomes.

Without replacement, branch probabilities change because the composition of the remaining population changes.

Independence

Events $A$ and $B$ are independent when knowing one does not change the probability of the other:

when $P(B)>0$. An equivalent test is

Independence must be justified, not assumed from events sounding unrelated.

Mutually exclusive is different

Mutually exclusive events cannot occur together, so $P(A\cap B)=0$. If both have positive probability, learning that $B$ occurred makes $A$ impossible; they are not independent.

Independence means no influence on probability, while mutual exclusivity means no overlap.

Bayes' rule

Bayes' rule reverses a condition:

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.

Quick self-check

• What event is known and therefore becomes the denominator?
• What intersection satisfies both the target and condition?
• Does a table or tree use consistent totals?
• Should path probabilities multiply or disjoint cases add?
• Are independence and mutual exclusivity being distinguished?
• If reversing a condition, have base rates been included?

Related topics

• Probability
• Tree Diagrams
• Bayes' Theorem
• Binomial Distribution