3.1 Basic Concepts of Probability and Counting

Probability Experiments

Probability is the measure of how likely an event is to occur. Probability experiments are actions or trials that produce specific results, called outcomes. The set of all possible outcomes is the sample space, and an event is any subset of the sample space.

• Probability Experiment: An action or process that leads to well-defined results (outcomes).

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

• Sample Space (S): The set of all possible outcomes.

• Event (E): Any subset of the sample space; may consist of one or more outcomes.

Example: Rolling a six-sided die:

• Sample Space: {1, 2, 3, 4, 5, 6}

• Event: Rolling an even number, {2, 4, 6}

• Outcome: Rolling a 2, {2}

Identifying Sample Spaces and Events

Sample spaces can be constructed for various experiments, often using tree diagrams to visualize all possible outcomes.

• Tree Diagram: A graphical representation showing all possible outcomes of a probability experiment.

Example: Surveying blood types (O, A, B, AB) and Rh factor (positive or negative) yields 8 possible outcomes:

• O positive, O negative, A positive, A negative, B positive, B negative, AB positive, AB negative

Simple and Compound Events

A simple event consists of a single outcome. An event with more than one outcome is a compound event.

• Simple Event: Selecting a specific defective part from a batch (only one outcome).

• Compound Event: Rolling at least a 4 on a die (outcomes: 4, 5, 6).

The Fundamental Counting Principle

The Fundamental Counting Principle is used to determine the number of ways multiple events can occur in sequence.

• If one event can occur in m ways and a second event in n ways, the total number of ways both can occur is m × n.

• This principle extends to any number of events: multiply the number of ways each event can occur.

Example: Choosing a car:

• Manufacturers: 3 (Ford, GM, Honda)

• Car sizes: 2 (compact, midsize)

• Colors: 4 (white, red, black, green)

• Total combinations:

Example: Car security code (4 digits, 0-9):

• Digits not repeated:

• Digits can repeat:

• First digit not 0 or 1, digits can repeat:

Types of Probability

There are three main types of probability:

• Classical (Theoretical) Probability: Used when all outcomes are equally likely.

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

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

The probability of event E is denoted as P(E).

• Classical Probability Formula:

• Empirical Probability Formula:

• Subjective Probability: No formal calculation; based on judgment.

Example: Rolling a six-sided die:

• P(rolling a 3):

• P(rolling a 7): $0$ (impossible event)

• P(rolling less than 5):

Empirical Probability and the Law of Large Numbers

Empirical probability is calculated from observed data. As the number of trials increases, empirical probability approaches theoretical probability, according to the Law of Large Numbers.

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

Example: Survey of 1502 adults on book reading habits:

P(next adult read only print books):

Range of Probabilities Rule

The probability of any event E is always between 0 and 1, inclusive:

• P(E) = 0: Impossible event

• P(E) = 1: Certain event

• P(E) = 0.5: Even chance

Events with probability ≤ 0.05 are considered unusual.

Complementary Events

The complement of event E (denoted E') consists of all outcomes not in E. The sum of the probabilities of an event and its complement is 1:

Thus,

Example: If P(user is 25-34 years old) = 0.255, then P(user is not 25-34 years old) = 1 - 0.255 = 0.745.

Probability Applications: Tree Diagrams and Counting

Tree diagrams can be used to visualize all possible outcomes and calculate probabilities for compound events.

• Example: Tossing a coin and spinning a spinner with 8 numbers:

• Total outcomes:

• P(tail and odd number):

• P(head or number > 3):

Summary Table: Types of Probability

Key Formulas

• Classical Probability:

• Empirical Probability:

• Complement:

• Counting Principle: For k events,

Additional info:

• Probability is foundational for inferential statistics, which uses sample data to make generalizations about populations.

• Tree diagrams and the Fundamental Counting Principle are essential tools for enumerating possible outcomes in complex experiments.