What is conditional probability, and how does it differ from joint probability?
Conditional probability P(A|B) is the probability of A given that B has already occurred, computed as P(A and B) / P(B). It narrows the sample space to B, whereas joint probability P(A and B) lives in the full, unrestricted space.
How to think about it
Conditional probability is the foundation of almost every probabilistic model. Get the definition and the ratio formula exactly right before building toward Bayes or independence.
Definition and formula
Given two events A and B with P(B) > 0:
P(A | B) = P(A ∩ B) / P(B)Reading left-to-right: the probability that A occurs, restricted to the world where B already occurred. Dividing by P(B) rescales so the conditional probabilities still sum to 1 over the reduced sample space.
Worked numeric example
A deck of 52 cards. Let A = “card is a King” and B = “card is a face card” (J, Q, K — 12 total).
P(A ∩ B)= 4/52 (4 Kings, all of which are face cards)P(B)= 12/52P(A | B)= (4/52) / (12/52) = 4/12 = 1/3
Once you know the card is a face card, exactly one-third of those are Kings — intuitive and verifiable by listing the Jack, Queen, and King of one suit.
Joint vs conditional at a glance
| Denominator | Space | |
|---|---|---|
P(A ∩ B) | Total outcomes | Full sample space |
P(A | B) | Outcomes in B | Reduced to B |
The joint probability can never exceed either marginal; the conditional probability can be larger or smaller than P(A) depending on whether B is evidence for or against A.