Conditional Probability
P(A given B) is the chance of A given that B happened.
About this topic
Conditional probability P(A given B) is the probability of A given that B has occurred: P(A given B) = P(A and B)/P(B). Knowing B shrinks the sample space to B alone.
It explains why drawing a second card changes the odds after the first is removed, and why a positive medical test does not always mean disease. Example: a bag has 3 red and 2 blue marbles. The chance that the second marble is red given that the first was red is 2/4 = 1/2, since only 4 marbles remain, 2 of them red. A table or tree diagram makes such calculations easy by showing which part of the whole you are focusing on.
Medical example: if a disease affects 1% of people and a test is 95% accurate at both detecting and ruling out, then of 10,000 people, 100 are sick and 95 test positive, while of the 9,900 healthy people about 495 test positive wrongly. A positive result therefore means only about 95/(95 + 495), roughly 16%, chance of the disease.
Conditional probability is vital in diagnosis, spam filters and risk analysis. Remember that P(A given B) is not the same as P(B given A); confusing them is a very common error.
Key ideas
- Restrict to outcomes where B happened
- P(A and B) = P(A given B) × P(B)
- Venn diagrams help
Quick check
Use cases: where Conditional Probability helps
- Homework: review the key idea behind Conditional Probability before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Conditional Probability without notes.
- Visual learning: play with the interactive lab to see how Conditional Probability changes when you adjust its inputs.
- Teaching a friend: explain Conditional Probability in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Conditional Probability?
P(A given B) is the chance of A given that B happened.
What will I learn in Conditional Probability?
Restrict to outcomes where B happened. P(A and B) = P(A given B) × P(B). Venn diagrams help.
Is Conditional Probability part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Conditional Probability lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Conditional Probability?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.