MATH 130: Module 0 Fundamentals

Interactive Review: Axioms, Inclusion-Exclusion, Bayes, & Independence

The 3 Axioms

  • 1

    Non-negativity

    For any event \(E\), \(0 \le P(E) \le 1\).

  • 2

    Normalization

    The probability of the sample space \(S\) is \(P(S) = 1\).

  • 3

    Additivity

    If \(E_1, E_2, \dots\) are disjoint (mutually exclusive), then \(P(\cup E_i) = \sum P(E_i)\).

Axiom Checker

Which of the following scenarios violate the axioms?

The Principle

Used when events are not disjoint.

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

For three events:

P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
- [P(A ∩ B) + P(A ∩ C) + P(B ∩ C)]
+ P(A ∩ B ∩ C)

Interactive Calculator

P(A ∪ B) = 0.8
A B A∩B

Bayes' Explorer

Adjust the parameters to see how the geometry changes.

0.30
0.90
0.20
Posterior Probability \(P(A|B)\)
--%
(Target / Total Colored)

The Formula

\( P(A|B) = \frac{\text{Dark Blue Area}}{\text{Total Blue Area}} \)
\( = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|A^c)P(A^c)} \)

Geometric Intuition (Area Model)

Imagine the entire square is the Sample Space (S).
1. We split it vertically into A (Left) and Aᶜ (Right).
2. We fill the bottom of each strip to represent B occurring.
3. P(A|B) is: "Given we are in the colored region (B), what fraction is Dark Blue (A)?"

A Aᶜ A ∩ B Aᶜ ∩ B
True Positives (A ∩ B)
False Positives (Aᶜ ∩ B)

Definition

Events \(A\) and \(B\) are independent if knowing one gives no information about the other.

Mathematical Definition

\( P(A \cap B) = P(A)P(B) \)

Equivalent Condition

\( P(A|B) = P(A) \)

Note: Disjoint events (with non-zero probability) are never independent! If A happens, B cannot happen, so knowing A gives you perfect information about B.

Independence Checker

Enter probabilities to check if events are independent.