Conditional probability is one of 2 conditional probability formulas in the probability section of this library, and it is used at high school · ap · university level.
Why conditional probability works
Conditioning on B throws away every outcome in which B did not happen, so B itself becomes the new universe. The joint probability is the part of that universe where A also happens, and dividing by the probability of B rescales it so the new universe totals 1 again.
What each symbol means
$A\cap B$ is the joint event.
Conditional probability: when it holds
$P(B)>0$.
When it stops applying
The condition must be an event that can actually happen, since dividing by a probability of zero is undefined. Conditioning on something impossible gives no new universe to work in, and there is nothing to rescale.
Conditional probability: a worked example
If $P(A\cap B)=.15$ and $P(B)=.3$, then $P(A\mid B)=.5$.
The mistake to avoid
What people do: Reversing the two events and using the chance of B given A in place of A given B.
Why it goes wrong: These are usually very different numbers. A screening test may catch 99 percent of people who have a disease, while only a small fraction of people who test positive actually have it.
Do this instead: Write the given event after the bar first, decide what universe that creates, and only then ask how much of it satisfies the other event.
Conditional probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A\cap B$ is the joint event.
- Check the conditions before substituting. $P(B)>0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Conditional probability
- Level
- High school · AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where conditional probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about conditional probability
Is the chance of A given B the same as the chance of B given A?
Almost never. They are equal only in the special case where the two events have the same probability, so treating them as interchangeable is one of the biggest sources of error in probability.
What happens when the two events are independent?
The condition tells you nothing, so the answer collapses back to the plain probability of A. That is exactly what independence means, written in conditional form.
How do I read this off a two-way table?
Restrict yourself to the row or column named by the condition, then divide the cell you want by that line's total. Conditioning is the same as switching from grand totals to row totals.
Can conditioning make an event more likely than it was?
Yes. Learning that a card is a face card raises the chance it is a king from 1 in 13 to 1 in 3, because the condition removed many outcomes where it could not be one.