Section 1.3: Conditional Probability
Learn how to calculate probabilities given that another event has occurred.
Conditional Probability Definition
The conditional probability of event A given that event B has occurred is:
provided P(B) > 0
Intuition
Given that B occurred, we're now working in a reduced sample space. P(A|B) asks: "Of all the ways B can happen, how many also include A?"
The Multiplication Rule
Rearranging the conditional probability formula gives us the Multiplication Rule:
Start with B, then A given B
Start with A, then B given A
Extended Multiplication Rule
P(A₁ ∩ A₂ ∩ A₃) = P(A₁) × P(A₂|A₁) × P(A₃|A₁ ∩ A₂)
Useful for drawing without replacement: each draw conditions on previous draws.
Tree Diagrams
Tree diagrams visually represent sequential events and their probabilities:
How to Use
- • Each branch represents a possible outcome
- • Branch probabilities are conditional on the path taken
- • Multiply along a path for joint probability
- • Add paths for total probability of an outcome
Rules
- • Branches from same node sum to 1
- • All leaf probabilities sum to 1
- • Each leaf represents a unique outcome
Classic Example: Drawing Without Replacement
Problem: An urn contains 5 red and 3 blue balls. Draw 2 balls without replacement. What is P(both red)?
Using Multiplication Rule:
P(R₁ ∩ R₂) = P(R₁) × P(R₂|R₁)
= (5/8) × (4/7) = 20/56 = 5/14 ≈ 0.357
Why 4/7 for the second draw?
After drawing a red ball: 4 red + 3 blue = 7 balls remain. P(red | first was red) = 4/7. This is the essence of "without replacement."
Law of Total Probability
If B₁, B₂, ..., Bₖ form a partition of the sample space (mutually exclusive and exhaustive), then:
When to Use
When you can't calculate P(A) directly, but you CAN calculate P(A) given different scenarios. Sum over all scenarios, weighting by scenario probability.