01 Key Concepts
Conditional Probability Notation
P(A|B) is read 'the probability of A given B' — the chance A happens, knowing B has already happened.
The Conditional Probability Formula
P(A|B) = P(A and B) / P(B), as long as P(B) is not zero.
Independent Events
Two events are independent if knowing one occurred doesn't change the probability of the other: P(A|B) = P(A).
Dependent Events
If P(A|B) does not equal P(A), the events are dependent — one affects the likelihood of the other.
Multiplication Rule
For any two events, P(A and B) = P(A) * P(B|A). If independent, this simplifies to P(A) * P(B).
02 Key Formulas
- P(A|B) = P(A and B) / P(B)
- P(A and B) = P(A) * P(B|A)
- Independent events: P(A and B) = P(A) * P(B)
03 Solved Examples
Example 1 A die is rolled. Find P(rolling a 4 | rolling an even number).
- Let A = rolling a 4, B = rolling an even number = {2,4,6}.
- A and B = {4}, so P(A and B) = 1/6.
- P(B) = 3/6.
- P(A|B) = P(A and B)/P(B) = (1/6)/(3/6).
Answer: P(A|B) = 1/3
Example 2 A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find the probability both are red.
- P(first red) = 5/8.
- Given the first is red, 4 red remain out of 7 total: P(second red | first red) = 4/7.
- Multiply: (5/8) * (4/7).
Answer: 20/56 = 5/14
Example 3 In a class, P(likes math) = 0.6, P(likes science) = 0.5, and P(likes both) = 0.3. Are liking math and liking science independent?
- Check if P(math and science) = P(math) * P(science).
- P(math) * P(science) = 0.6 * 0.5 = 0.30.
- This matches the given P(both) = 0.3.
Answer: Yes, the events are independent
04 Practice Questions
1A die is rolled. Find P(rolling a 2 | rolling an even number).
1/3
2A bag has 4 red and 6 blue balls. Find P(drawing 2 red without replacement).
(4/10)*(3/9) = 2/15
3If P(A)=0.4, P(B)=0.5, and events are independent, find P(A and B).
0.2
4If P(A and B) = 0.2 and P(B) = 0.4, find P(A|B).
0.5
5If P(A|B) = P(A), are A and B independent or dependent?
Independent
📄 Conditional Probability — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.