What Is Conditional Probability?
Grade 6When information changes the probability
Sometimes you're told that something has already happened — and that changes the probability of something else. That's conditional probability.
The notation P(A | B) is read as "the probability of A given B". It means: knowing that B has happened, what's the chance of A?
The key move is that your sample space has shrunk. You're no longer looking at all outcomes — only the ones where B has occurred.
The formula
In words: out of all the times B happens, how many of those also have A? The denominator is P(B) because you're restricting yourself to the world where B occurred.
Worked example — from a table
The shrinking sample space
Try this one
The two-way table shows whether \( 40 \) students walk or cycle to school.
| Walk | Cycle | Total | |
|---|---|---|---|
| Boys | 9 | 15 | 24 |
| Girls | 11 | 5 | 16 |
| Total | 20 | 20 | 40 |
A student is chosen at random.
Given that the student is a boy, find the probability that he cycles to school. Give your answer as a fraction in its simplest form. [2]
This is conditional probability: the word 'given' tells you to look only within one group.
Ask yourself: 'which row or column should I restrict my attention to, and what is the total of just that group?' The denominator is the size of the group you are told about, not the whole table.
