Joint Probability — Definition, Formula & Examples

Joint probability is the probability that two (or more) events happen together, such as a randomly chosen student taking both Calculus and Physics. You write it as

P(A \cap B)

P(A, B)

For events $A$ and $B$ in a sample space $S$ , the joint probability is $P(A \cap B)$ , the probability of the event consisting of all outcomes in both $A$ and $B$ . For discrete random variables $X$ and $Y$ , the joint probability distribution is the function $p(x, y) = P(X = x,\, Y = y)$ , which satisfies $p(x,y) \ge 0$ and $\sum_x \sum_y p(x, y) = 1$ .

Key Formula

P(A \cap B) = P(A)\,P(B \mid A) = P(B)\,P(A \mid B)

Where:

$P(A \cap B)$ = Joint probability that both A and B occur
$P(A)$ = Probability that event A occurs
$P(B \mid A)$ = Conditional probability of B given that A has occurred
$P(A \mid B)$ = Conditional probability of A given that B has occurred

Worked Example

Problem: In a group of 100 students, 40 take Calculus. Of those 40, 30 also take Physics. Another 20 students take Physics but not Calculus. You pick one student at random. Find the joint probability that the student takes both Calculus and Physics.

Step 1: Count the students in the intersection (both courses) and divide by the total. This is how you read a joint probability straight from a two-way table.

P(C \cap P) = \frac{30}{100} = 0.3

Step 2: Check with the multiplication rule. First find

P(C) = \frac{40}{100}

and

P(P \mid C) = \frac{30}{40}

, then multiply.

P(C)\,P(P \mid C) = \frac{40}{100} \cdot \frac{30}{40} = 0.4 \cdot 0.75 = 0.3

Answer: The joint probability that a randomly chosen student takes both Calculus and Physics is

P(C \cap P) = 0.3

Why It Matters

Joint probabilities are the numerator in conditional probability and Bayes' theorem. That is how you compute the chance a patient has a disease given a positive test. In statistics and machine learning courses, joint distributions of several variables are the starting point for covariance, correlation, and probabilistic models.

Common Mistakes

Mistake: Multiplying

P(A)\,P(B)

to get the joint probability for any two events. Here that gives

P(C)\,P(P) = 0.4 \cdot 0.5 = 0.2

Correction: The shortcut

P(A \cap B) = P(A)\,P(B)

works only when

A

and

B

are independent. Taking Calculus and taking Physics are not independent here, so the correct value is

0.3

. In general, use

P(A)\,P(B \mid A)

Related Terms

Probability Distribution — the broader concept this term builds on
Probability — the broader concept this term builds on
Conditional Probability — Defined as joint probability divided by a marginal
Bayes' Theorem — Rewrites joint probability to reverse a conditional
Addition Rule — Subtracts the joint probability to find P(A or B)
Event — Joint probability concerns the intersection of events
Complement of an Event — Fills the remaining cells of a two-way table
Binomial Probability Formula — Built from joint probabilities of independent trials