Section 5.4

Conditional Probability & The General Multiplication Rule

How does "given" information change the odds? Learn to calculate probabilities when the sample space shrinks and how to handle dependent events formally.

Conditional Probability

Shrinking the Sample Space

Conditional Probability is the probability that event F occurs, given that event E has already occurred. This "given" condition effectively filters the universe, creating a smaller sample space.

The Formula

Read as: "Probability of F given E"

Contingency Table Explorer

	Smoker	Non-Smoker	Row Total
Male	120	280	400
Female	80	320	400
Col Total	200	600	800

Association Check

Male

Smoker: 30.0%

Female

Smoker: 20.0%

The distributions are different — suggests an ASSOCIATION (Dependent).

Conditional Distributions

Standard Probability

"Prob of picking a King from a deck?"

4 / 52

Denominator is Total Outcomes

Conditional Probability

"Prob of King GIVEN it's a Face Card?"

4 / 12

Denominator is "Face Cards" only

The General Multiplication Rule

When events are dependent, we cannot simply use $P(E) \cdot P(F)$. We must adjust the second probability to account for the first event occurring.

LogicLens Intuition: The "Natural" Flow

Think of this as a story unfolding in time:

1. First, Event E must happen: P(E).
2. Since E happened, our world has changed. Now Event F must happen in this new world: P(F|E).

Multiplying them gives the probability of the entire story sequence.

Tree Diagrams

Tree diagrams are perfect for visualizing multi-stage experiments, especially when probabilities change (dependent events).

1Each branch represents a possible outcome of a stage.
2Probabilities on secondary branches are always conditional ().
3Multiply along the branches to find the probability of the final path ().

Interactive Tree Diagrams

Selecting 2 circuits from a batch of 100 (5 Defective, 95 Good) WITHOUT replacement.

Interactive Controls

Total Circuits

Defective Count

Start

95/100

Good (G)

5/100

Defective (D)

Good (GG)

0.950 × 0.949

= 0.902

Defective (GD)

0.950 × 0.051

= 0.048

Reject

Good (DG)

0.050 × 0.960

= 0.048

Reject

Defective (DD)

0.050 × 0.040

= 0.002

Reject

Independence Revisited (The Test)

We intuitively know "coin flips" are independent. But how do we prove it mathematically? We check if the "given" information actually changes anything.

Formal Definition

Two events E and F are independent if and only if:

"If knowing that E occurred does NOT change the probability of F, then E has no influence on F. They are independent."

LogicLens Practice

Conditional Probability & The General Multiplication Rule