Probability

Basic Concepts of Probability and Counting

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring in a random experiment. Understanding probability involves identifying the sample space, events, and outcomes, and applying counting principles to determine possible results.

• Probability Experiment: An action or trial that produces specific results (counts, measurements, or responses).

• Outcome: The result of a single trial in a probability experiment.

• Sample Space: The set of all possible outcomes of a probability experiment.

• Event: A subset of the sample space, consisting of one or more outcomes.

Examples of Probability Experiments

• Basketball Shot: Outcomes are 0, 1, 2, or 3 points.

• Traffic Light: Outcomes are green, yellow, or red.

Identifying the Sample Space

For a survey asking blood types (O, A, B, AB) and Rh factor (positive or negative), the sample space consists of eight possible outcomes, which can be visualized using a tree diagram.

Simple Events

A simple event is an event that consists of a single outcome. If an event includes more than one outcome, it is not simple.

• Example: Selecting a specific defective machine part is a simple event (only one outcome).

• Example: Rolling at least a 4 on a six-sided die is not a simple event (three outcomes: 4, 5, 6).

Counting Skills and the Fundamental Counting Principle

Tree diagrams are useful for listing outcomes when possibilities are few, but for larger sets, the Fundamental Counting Principle is used to efficiently determine the number of possible outcomes.

• Fundamental Counting Principle: If one event can occur in m ways and another in n ways, the two events together can occur in m × n ways.

Example: Car Selection

• Three manufacturers, two car sizes, four colors: Total ways = 3 × 2 × 4 = 24.

Example: Access Codes

• Four digits, each 0–9, no repetition: 10 × 9 × 8 × 7 = 5040 possible codes.

• With repetition: 10 × 10 × 10 × 10 = 10,000 possible codes.

• First digit not 0 or 1: 8 × 10 × 10 × 10 = 8,000 possible codes.

Example: License Plates in Ohio

• Three letters (A–Z), four digits (0–9), no repetition: 26 × 25 × 24 × 10 × 9 × 8 × 7 = 175,760,000 possible plates.

Types of Probability

Probability can be classified into three main types: classical, empirical, and subjective.

• Classical (Theoretical) Probability: Each outcome in the sample space is equally likely. Formula: where n(E) is the number of outcomes in event E, and n(S) is the number of outcomes in the sample space.

• Empirical (Statistical) Probability: Based on observed data from experiments. Formula: where f is the frequency of the event, and n is the total number of trials.

• Subjective Probability: Based on intuition, educated guesses, or estimates.

Example: Finding Classical Probabilities

• Rolling a 3 on a six-sided die:

• Rolling a number less than 5:

Example: Finding Empirical Probabilities

• Survey of U.S. adults reading books: Probability that the next adult read only print books =

Law of Large Numbers

The law of large numbers states that as a probability experiment is repeated many times, the empirical probability approaches the theoretical probability.

Subjective Probability Examples

• A doctor estimates a 90% chance of recovery.

• A sports analyst predicts a 70% chance of winning.

• An investor estimates a 40% chance of market rise.

Range of Probabilities Rule

The probability of any event E is between 0 and 1, inclusive. This range can be interpreted as follows:

• 0: Impossible

• 0.25: Unlikely

• 0.5: Even chance

• 0.75: Likely

• 1: Certain

Complementary Events

The complement of event E is the set of all outcomes in the sample space not included in E. The probability of the complement is:

• Formula:

Example: Probability of the Complement

• If 765 out of 3000 users are 25–34 years old, then the probability that a randomly selected user is not 25–34 years old is

Using the Fundamental Counting Principle: Social Security Numbers

A Social Security Number (SSN) consists of nine digits, each from 0 to 9, and digits can be repeated. The total number of possible SSNs is:

• Formula: possible SSNs.

• Probability of randomly generating your SSN:

Summary Table: Types of Probability

Key Formulas

• Classical Probability:

• Empirical Probability:

• Complement:

Additional info: Academic context and examples were expanded for clarity and completeness. All images included are directly relevant to the adjacent explanations.