P(A\mid B)=\frac{P(A\cap B)}{P(B)}Variables
- P(A|B): probability of A given B
- P(A ∩ B): probability both occur
- P(B): probability of the condition
How to use this formula
Calculates the probability of A when B is known to have occurred.
Important notes
- P(B) must be nonzero.
- Conditional probability changes the relevant sample space to outcomes in B.
Quick example
If 20% of people meet both conditions and 50% meet B, then P(A|B) = 0.4.
Applicability, worked calculation, and verification
Domain and applicability
The conditioning event must have nonzero probability; all counts or probabilities must refer to the same population and sampling frame.
Units
- probabilities are dimensionless
- counts use the same population base
Assumptions and domain checks
- P(B) must be nonzero.
- For the Conditional Probability, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- The event model, conditioning information, independence assumptions, and probability range must match the problem.
Do not use this formula when
- Do not use Conditional Probability until the sample, event, distribution, independence assumptions, and parameter convention match the problem.
Boundary and special cases
- For Conditional Probability, check zero, negative, and extreme input values before relying on the result.
- When using Conditional Probability, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form P(A\mid B)=\frac{P(A\cap B)}{P(B)} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Among 18 students who study mathematics, 12 also study coding. P(coding|mathematics) = 12/18 = 2/3 ≈ 66.7%.
- Restrict the sample space to the 18 students satisfying the condition.
- Count the 12 students who satisfy both mathematics and coding.
- Divide the intersection count by the condition count: 12/18.
- Simplify to 2/3 and convert to approximately 66.7%.
Independent verification
Multiplying P(coding|mathematics) by P(mathematics) recovers the joint probability P(coding ∩ mathematics) for the same population.
Common mistakes
- When copying Conditional Probability, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not assume events are independent or mutually exclusive unless the problem states or proves that condition.
Continue the workflow
Use Conditional Probability in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
P(A\mid B)=\frac{P(A\cap B)}{P(B)}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Conditional Probability.
Verified against: OpenStax Introductory Statistics
Automated quality check: Passed the core formula indexing gate.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Conditional Probability used for?
Calculates the probability of A when B is known to have occurred.
Can I copy this formula as LaTeX?
Yes. Copy P(A\mid B)=\frac{P(A\cap B)}{P(B)} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.