Probability Fundamentals

A. Basic Concepts

Probability theory is the mathematical study of random phenomena and uncertainty. It provides a framework for quantifying the likelihood of events in experiments with uncertain outcomes.

• Experiment: Any procedure with an uncertain outcome. Example: Tossing a die.

• Sample Space (S): The set of all possible outcomes. Example: for a die toss.

• Event (E): Any subset of the sample space. Example: (even numbers on a die).

B. Set Operations

Set operations are used to describe relationships between events in probability.

• Union (A ∪ B): Outcomes in A or B.

• Intersection (A ∩ B): Outcomes in both A and B.

• Complement (Ac): Outcomes not in A.

• Mutually Exclusive: (no outcomes in common).

C. Definition of Probability

Probability assigns a numerical value to the likelihood of events, following certain axioms (Kolmogorov's axioms):

• 1. (the probability of the sample space is 1).

• 2. for all events .

• 3. If are disjoint, then:

D. Rules of Probability

Several rules help calculate probabilities for combined or related events.

• Complement Rule:

• Addition Rule:

• If A and B are disjoint:

E. Counting Methods

Counting methods are essential for determining the number of possible outcomes in probability problems.

• Multiplication Principle: If one event can occur in ways and another in ways, both together can occur in ways.

• Permutations: Number of ways to arrange objects:

• Combinations: Number of ways to choose objects from without regard to order:

F. Conditional Probability & Independence

Conditional probability measures the likelihood of an event given that another event has occurred. Independence describes events that do not affect each other's probabilities.

• Conditional Probability: , provided

• Independence: Events A and B are independent if

G. Total Probability & Bayes' Theorem

The total probability rule and Bayes' theorem are used to compute probabilities when events are partitioned or when updating beliefs based on new information.

• Total Probability: If partition , then

• Bayes' Theorem: