Probability Basics

Probability Fundamentals

Definition of Probability

Probability is a measure of the likelihood that an event will occur.

\[P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}}\]

Probability values: \(0 \leq P(A) \leq 1\)

Basic Rules

1. Complement Rule

\[P(A^c) = 1 - P(A)\]

Example: If probability of rain = 0.3, then probability of no rain = 1 - 0.3 = 0.7

2. Addition Rule

For mutually exclusive events: \[P(A \cup B) = P(A) + P(B)\]

For general events: \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

3. Multiplication Rule

For independent events: \[P(A \cap B) = P(A) \times P(B)\]

Conditional Probability

Probability that A occurs given that B has occurred: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

Example: Rolling a Die

• P(rolling a 6) = 1/6
• P(even number) = 3/6 = 1/2
• P(number greater than 4) = 2/6 = 1/3

Example: Flipping a Coin

• P(heads) = 1/2
• P(tails) = 1/2
• P(heads twice in a row) = 1/2 × 1/2 = 1/4