Probability · Quick Reference

Conditional Probability Cheat Sheet

Probability · Lesson 3/8

In one line: conditional probability answers the question: given that something has already happened, how likely is something else? It updates our probability estimate using new information.

Key Ideas

1Conditional Probability Notation. P(A|B) is read 'the probability of A given B' — the chance A happens, knowing B has already happened.

2The Conditional Probability Formula. P(A|B) = P(A and B) / P(B), as long as P(B) is not zero.

3Independent Events. Two events are independent if knowing one occurred doesn't change the probability of the other: P(A|B) = P(A).

4Dependent Events. If P(A|B) does not equal P(A), the events are dependent — one affects the likelihood of the other.

5Multiplication Rule. For any two events, P(A and B) = P(A) * P(B|A). If independent, this simplifies to P(A) * P(B).

Worked Examples

A die is rolled. Find P(rolling a 4 | rolling an even number).

P(A|B) = 1/3

A bag has 5 red and 3 blue balls. Two balls are drawn without replacement. Find the probability both are red.

20/56 = 5/14

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?

Yes, the events are independent

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)

Practice Yourself

A die is rolled. Find P(rolling a 2 | rolling an even number).

1/3

A bag has 4 red and 6 blue balls. Find P(drawing 2 red without replacement).

(4/10)*(3/9) = 2/15

If P(A)=0.4, P(B)=0.5, and events are independent, find P(A and B).

0.2