Chapter 3: Probability – Study Guide

Probability

Basic Concepts of Probability & Counting

Probability is a measure of how likely an event is to occur. It is foundational in statistics and is used to quantify uncertainty. Key concepts include experiments, outcomes, sample spaces, events, and simple event definitions.

• Experiment: A process that leads to well-defined outcomes.

• Outcome: A possible result of an experiment.

• Sample Space: The set of all possible outcomes.

• Event: Any subset of a sample space.

• Simple Event: An event with a single outcome.

Fundamental Counting Principle: If one event can occur in m ways and a second event can occur in n ways, then the two events can occur in m × n ways.

Types of Probability

• Classical Probability: Based on the assumption that all outcomes are equally likely.

• Empirical (Statistical) Probability: Based on observations from experiments.

• Subjective Probability: Based on personal judgment or experience.

Law of Large Numbers: As an experiment is repeated many times, the empirical probability of an event approaches its theoretical probability.

Complement Rule: The probability that event A does not occur is given by:

$P(A') = 1 - P(A)$

Conditional Probability & Multiplication Rule

Conditional probability is the probability of an event occurring given that another event has already occurred.

• Conditional Probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Multiplication Rule (for independent events): $P(A \cap B) = P(A) \times P(B)$

• Multiplication Rule (for dependent events): $P(A \cap B) = P(A) \times P(B|A)$

Independent Events: The occurrence of one event does not affect the probability of the other.

Dependent Events: The occurrence of one event affects the probability of the other.

Addition Rule

The addition rule is used to find the probability that at least one of two events occurs.

• Mutually Exclusive Events: Events that cannot occur at the same time.

• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• For mutually exclusive events: $P(A \cup B) = P(A) + P(B)$

Counting Techniques

Counting techniques are used to determine the number of possible outcomes in a sample space.

• Permutations: Arrangements of objects where order matters. $nPr = \frac{n!}{(n - r)!}$

• Combinations: Selections of objects where order does not matter. $nCr = \frac{n!}{r!(n - r)!}$

• Long Counting Problems: Use the fundamental counting principle to solve complex probability problems.

Example: If you have 3 shirts and 2 pants, the number of possible outfits is $3 \times 2 = 6$.