A conditional probability asks how likely A is given that B has happened.
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). The condition shrinks the world to just B.
From a table
Condition on a row or column by using that total as the denominator instead of the grand total.
Order matters
P(A given B) and P(B given A) are different questions with different denominators, and usually different answers.
A test for independence
If P(A given B) equals P(A), the condition changed nothing and the events are independent.
Why it matters
Most people who are ill test positive, yet most people testing positive may still be well when the illness is rare. Swapping the two is a costly error.
Conditioning changes the denominator
P(A | B) restricts attention to outcomes where B occurred, so you divide by P(B) rather than by the whole. That change of denominator is the entire content of conditioning.
The order is not reversible
P(A | B) and P(B | A) are generally very different. The chance of a positive test given illness is not the chance of illness given a positive test, and confusing them has real consequences.
Base rates dominate rare events
A highly accurate test for a rare condition still produces mostly false positives, because there are so many more unaffected people to be wrong about. Ignoring the base rate is the classic failure.
Tree diagrams organise the conditioning
Each branch carries a conditional probability and multiplying along a path gives the joint probability. Drawing the tree turns a confusing problem into arithmetic.
Step 2: Try It Yourself
Tap and try it out.
Group A
Group B
Group A has the most. It has 4 more than Group B.
Step 3: Watch an Example
One step at a time.
Watch Sana Condition on a Row
Of 100 students, 60 play sport, and 45 of those also play music.
- Step 1
The condition is playing sport, so the world shrinks from 100 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) = 0.12, P(B) = 0.4. P(A given B)?
3 of 8
18 of 24 in a group have a trait. P(trait given the group)?
4 of 8
P(A given B) = P(A). Independent? 1 yes, 0 no.
5 of 8
A die shows an even number. P(it is a 6), to three decimal places?
6 of 8
Is P(A given B) always equal to P(B given A)? 1 yes, 0 no.
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) = 0.35, P(B) = 0.7. P(A given B)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
P(A and B) = 0.18, 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.
- If conditioning changes nothing, the events are independent.