Conditional probability formula: P(A | B) = P(A ∩ B) / P(B)
The conditional probability formula is P(A | B) = P(A ∩ B) / P(B).
What each letter means
- \(P(A\mid B)\) the probability of A, given that B has happened
- \(P(A\cap B)\) the probability of A and B
- \(P(B)\) the probability of the event you know has happened
When to use it
When you are told something has already happened ('given that…'), so the sample space shrinks to B.
Worked example
Of 60 students, 24 study French and 9 study both French and Spanish. A student who studies French is chosen. Find the probability that they also study Spanish.
- \(P(S\mid F)=\dfrac{P(S\cap F)}{P(F)}=\dfrac{9/60}{24/60}=\dfrac{9}{24}\)
Answer: \(P(S\mid F)=0.375\)
Common mistake
Dividing by P(A) instead of P(B). Divide by the probability of the event after the bar.
On your course
| Course | In the exam |
|---|---|
| AA SL | In the IB formula booklet |
| AI SL | In the IB formula booklet |
| AA HL | In the IB formula booklet |
| AI HL | In the IB formula booklet |
From our own IB Maths formula sheets, in our words. The IB gives its booklet to schools, so we checked against our own copy: your teacher has the official one. Official: IB DP Mathematics page.
Practise and revise
- Practise AA HL conditional probability questions
- Practise AA SL probability questions
- Revise Conditional probability (AI SL)
- Revise Conditional probability (AA HL)
- Print AA SL one-page formula sheet
Questions
What is the conditional probability formula?
The conditional probability formula is P(A | B) = P(A ∩ B) / P(B). P(A| B): the probability of A, given that B has happened; P(A∩ B): the probability of A and B; P(B): the probability of the event you know has happened.
Is the conditional probability given in the exam?
AA SL: in the IB formula booklet. AI SL: in the IB formula booklet. AA HL: in the IB formula booklet. AI HL: in the IB formula booklet. This comes from our own IB Maths formula sheets; your teacher has the official booklet.
How can I tell if A and B are independent?
A and B are independent when P(A | B) = P(A), or equivalently P(A ∩ B) = P(A) × P(B).
Our own wording, examples and card, checked by IB Math Revision. Not produced or endorsed by the International Baccalaureate Organization.