Conditional Probability
Simple Explanation
Conditional probability, P(A|B), is the probability of A happening given that B is already known to have happened. Knowing B occurred narrows the sample space down to just B's outcomes, and P(A|B) = P(A∩B) / P(B) measures A's share of that narrowed space.
Why Do We Need It?
Real information often arrives partially — conditional probability is how you correctly update a probability once you learn that something else has already happened.
Formula
Conditional Probability
P(A | B) = P(A ∩ B) / P(B) (P(B) ≠ 0)
The probability that A happens, given that B is already known to have happened — restricting the sample space down to just the outcomes where B occurs.
- P(A | B)
- — the probability of A, given that B has occurred
- P(A ∩ B)
- — the probability that both A and B happen
- P(B)
- — the probability that B happens
When to use it: Whenever extra information (that some other event B has already happened) changes the sample space you should be considering.
Worked Example
Find a conditional probability
A card is drawn from a standard deck. Given that it is a face card (J, Q, K), find the probability it is a king.
Why Does This Work?
Once B is known to have happened, only outcomes within B remain relevant — dividing by P(B) re-normalizes A∩B's probability as a fraction of this new, smaller sample space, rather than the original full sample space.
Real-Life Example
Medical test results
A doctor wants to know the probability a patient has a disease, given that their test result came back positive.
P(disease | positive test) is a conditional probability — using known information (a positive test) to update the probability of the actual condition (having the disease), rather than the plain unconditional disease rate.
Practice
P(A∩B) = 0.15 and P(B) = 0.3. Find P(A|B).
HardCommon mistake
Confusing P(A|B) with P(B|A) — these are generally different values; "probability of A given B" is not the same as "probability of B given A" unless P(A) happens to equal P(B).
Quick Review
- P(A|B) = P(A∩B) / P(B) — the probability of A within the reduced sample space where B is true.
- Conditional probability updates a probability based on new information.
- P(A|B) and P(B|A) are generally different — do not mix them up.