Chapter 5: Probability Theory

Section 5.1: General Probability Rules

Recall from section 4.2:

1. The probability, P(A) of any event A is 0 ≤ P(A) ≤ 1.
2. P(S) = 1 or P(Ω) = 1.
3. Complement rule: P(A^c) = 1 – P(A). In other words, P(A does not occur) = 1 – P(A).
4. Two events A and B are called disjoint if they have no outcomes in common. If A and B are disjoint,
   then P(A È B) = P(A) + P(B). In other words, P(A or B) = P(A) + P(B).
   In general, if A₁, A₂, ¼, A_k are disjoint, then P(A₁ È A₂ È · · · È A_k) = P(A₁) + P(A₂) + · · · + P(A_k)

Independence and the multiplication rule

Venn diagrams: a method using pictures to show whether two events are disjoint. Click here for practice using Venn diagrams.

Independent events: Two events are independent if the outcome of one event does NOT influence the outcome of the other event. Examples include: tossing two coins, rolling two die, choosing two cards from a deck of cards (replacing the first card before choosing the second one).

5. Multiplication Rule for independent events:
   If two events A and B are independent, then P(A and B) = P(A)P(B).
   In general, if events A, B, C, …, Z are independent, then P(A and B and … Z) = P(A)P(B)···P(Z).

Applying the multiplication rule

If A and B are independent, then A and B^c are independent!

The general addition rule

6. General Addition Rule for any two events:
   For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B).
   If events A and B are disjoint, then P(A and B) = 0.
   If events A and B are NOT disjoint, then P(A and B) is NOT = 0.