Resource:
- Book Probability by Jim Pitman, Springer, Page 33
- C. to, “measure of likelihood of an event when another event is known to have occurred,” Wikipedia.org, Mar. 06, 2001. https://en.wikipedia.org/wiki/Conditional_probability (accessed Mar. 04, 2026).
What is Conditional Probability?
Conditional Probability
Conditional probability is the likelihood of an event occurring, given that another event has already happened. It allows us to update our understanding of an outcome based on new information or specific “conditions.”
The Formula
P(A|B) =
P(A ∩ B)
P(B)
- • P(A|B): The probability of event A occurring given B has occurred.
- • P(A ∩ B): The probability of both A and B occurring (the intersection).
- • P(B): The probability of the condition B occurring.
A Simple Example: Rolling a Die
How new information changes the outcome.
A
Rolling a 4
B
Rolling an Even {2, 4, 6}
Standard World
Without info, the sample space is:
Probability P(A) = 1/6 (16.7%)
{1, 2, 3, 4, 5, 6}
Probability P(A) = 1/6 (16.7%)
Conditional Probability
Define Event A (Target) and Condition B (Given)
MATH CALCULATION STEPS:
1. Total Sample Space (S) = {1, 2, 3, 4, 5, 6}
2. Condition B = {1, 2, 3, 4, 5, 6}
3. A ∩ B (Is Target in B?) = YES
P(A|B) = 1 / 6 = 0.167
16.7%
Joint Probability Simulator
Find $P(A|B)$ where $A$ is a number and $B$ is the sum.
Calculation Log:
B (Sum=7): (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
Intersection (contains 3): (3,4), (4,3)
P(A|B) = 2 / 6 = 0.333
33.3%