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
P(A | B) = P(A ∩ B) ÷ P(B)
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
EXAMPLE
A group of 50 people were asked about tea and coffee. 30 drink tea (T), 25 drink coffee (C), 12 drink both. A person is chosen at random. Given that they drink coffee, find the probability they also drink tea.
P(T ∩ C) = 12/50. P(C) = 25/50.
P(T | C) = P(T ∩ C) ÷ P(C) = (12/50) ÷ (25/50) = 12/25 = 12/25.
In plain English: of the 25 coffee drinkers, 12 also drink tea — same answer, just counting directly.
The shrinking sample space
You can always answer a conditional probability question by restricting your focus: "how many people are in group B?" becomes your new total, and you count how many of those are also in A. No formula needed if you can see the numbers directly.
