Conditional probability
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]
Probability
Conditional Probability
Understand
If we know that event \(B\) has occurred, the sample space effectively shrinks to \(B\). The probability of \(A\) given \(B\) is
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)},\qquad P(B)\ne0.\]
For equally likely outcomes this is simply \(\dfrac{n(A\cap B)}{n(B)}\): count the outcomes of \(B\), then count those that are also in \(A\).
Example. Two dice are thrown. Given that the first shows an even number, the probability that the total is 8 is \(\frac{3}{18}=\frac16\) (the pairs \((2,6),(4,4),(6,2)\) among the 18 outcomes with an even first die).
Properties
- \(P(S\mid F)=P(F\mid F)=1\).
- \(P(A\cup B\mid F)=P(A\mid F)+P(B\mid F)-P(A\cap B\mid F)\).
- \(P(A'\mid F)=1-P(A\mid F)\).
Key Concepts
- Restrict attention to the given event.
- P(A | B) = P(A ∩ B)/P(B).
- P(A | B) and P(B | A) are usually different.
Formula Bank
Common Mistakes
Reversing the condition
\(P(A\mid B)\) divides by \(P(B)\); \(P(B\mid A)\) divides by \(P(A)\). They are equal only when \(P(A)=P(B)\).
Practice & Topic Test
Topic Practice
A fresh set of questions from this topic. Change answers freely – solutions appear after you submit.
Start PracticeTopic Summary
Conditional probability rescales probabilities to the event that is known to have happened.
Ready to practise this topic?
Create Student Account
Create a free account to take practice sessions and tests, see detailed explanations and track your progress.