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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(Ac) = 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, Ak are disjoint, then P(A1 È A2 È · · · È Ak) = P(A1) + P(A2) + · · · + P(Ak)

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).

  1. 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 Bc are independent!

The general addition rule

  1. 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.

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