A conditional probability asks how likely A is, given that B has already happened. It is written P(A given B).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The formula
P(A given B) = P(A and B) ÷ P(B). Knowing B has happened shrinks the world to just B.
A smaller denominator
The condition replaces the whole sample space with a slice of it. That is the entire idea in one sentence.
Order matters
P(A given B) and P(B given A) are usually different numbers. Most people who are ill test positive; most people who test positive may still be well.
Two-way tables
Condition on a row or column by using that row or column total as the denominator instead of the grand total.
A test for independence
If P(A given B) equals P(A), the condition changed nothing, and the events are independent.
Given that something already happened
P(A | B) restricts attention to the outcomes where B occurred, so the denominator shrinks to P(B). That change of denominator is the whole content of conditioning.
The order is not reversible
P(A | B) and P(B | A) are generally quite different. The probability that someone tests positive given they are ill is not the probability they are ill given a positive test. Swapping them is a serious and common error.
Base rates matter enormously
A test that is 99% accurate for a disease affecting 1 in 10,000 people still produces mostly false positives, because there are so many more healthy people to be wrong about. Ignoring the base rate is the classic medical statistics failure.
Tree diagrams organise conditioning
Each branch carries a conditional probability, and multiplying along a path gives the joint probability. Drawing the tree is what turns a confusing conditional problem into arithmetic.
Step 2: Try It Yourself
Tap and try it out.
Walk
Bus
Car
Walk has the most. It has 7 more than Car.
Step 3: Watch an Example
One step at a time.
Watch Elena Read a Two-Way Table
Of 100 students, 60 play sport. Of those 60, 45 also play music. Elena wants P(music given sport).
- Step 1
The condition is playing sport, so the world shrinks from 100 students to 60.
Step 4: Your Turn
Practice makes it stick.
The Condition
Problem 1 of 2
P(A and B) = 0.2 and P(B) = 0.5. What is P(A given B)?
The Table
Problem 2 of 2
Of 60 sport players, 45 play music. What is P(music given sport)?
Given That
1 of 8
P(A and B) = 0.3, P(B) = 0.6. P(A given B)?
2 of 8
P(A and B) = 0.1, P(B) = 0.4. P(A given B)?
3 of 8
20 of 50 in a group have a trait. P(trait given the group)?
4 of 8
P(A given B) = P(A) = 0.3. Are A and B independent? 1 yes, 0 no.
5 of 8
Is P(A given B) always equal to P(B given A)? 1 yes, 0 no.
6 of 8
A die shows an even number. What is P(it is a 6), as a decimal to three places?
7 of 8
Put the conditional probability method in order.
- 1Use that event as the new total.
- 2Count how many of those also satisfy the other event.
- 3Divide to get the conditional probability.
- 4Identify which event is the condition.
8 of 8
P(A and B) = 0.45, P(B) = 0.9. P(A given B)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
P(A and B) = 0.24, P(B) = 0.6. What is P(A given B)?
Question 2 of 2
What does conditioning on B do?
What You Learned
- P(A given B) = P(A and B) ÷ P(B).
- Conditioning replaces the sample space with a slice of it.
- P(A given B) and P(B given A) are different questions.