Q1. A die is thrown. Given the number is even, find the probability it is 2.
B = even = {2, 4, 6}, so P(B) = 3/6
A∩B = {2}, so P(A∩B) = 1/6
P(A|B) = (1/6)/(3/6) = 1/3
Probability — Class 12
Class 12 probability is all about conditions. Master conditional probability and Bayes' theorem, and compute basic probabilities live.
Conditional probabilityP(A|B) = P(A∩B)/P(B) — the probability of A given that B has happened.
Multiplication theoremP(A∩B) = P(A)·P(B|A). For independent events, P(A∩B) = P(A)·P(B).
Independent eventsEvents whose outcomes do not affect each other: P(A|B) = P(A).
Bayes' theoremReverses the condition: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).
Conditional probability calculator Interactive
Enter P(A∩B) and P(B) to get the conditional probability P(A|B).
P(A | B) = P(A∩B)/P(B)—
For independent events P(A|B) = P(A), so the condition makes no difference.
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Conditional probability | P(A|B) = P(A∩B)/P(B) |
| Multiplication | P(A∩B) = P(A)·P(B|A) |
| Independent events | P(A∩B) = P(A)·P(B) |
Common mistakes & exam wins
- Bayes' theorem reverses a condition — use it for 'given the effect, find the cause' problems.
- For independent events, multiply the probabilities: P(A∩B) = P(A)·P(B).
- Always identify the conditioning event (what is "given") first.
Frequently asked questions
What is conditional probability?
P(A|B) = P(A∩B)/P(B), the probability of A given that B has already occurred.
What is the multiplication theorem of probability?
P(A∩B) = P(A)·P(B|A); for independent events it becomes P(A)·P(B).
What is Bayes' theorem used for?
To find the probability of a cause given an observed effect, by reversing a conditional probability.