Probability

Basic Concepts of Probability and Counting

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

• 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: One or more outcomes; a subset of the sample space.

Example: Identifying the Sample Space

Suppose a survey asks for blood type (O, A, B, AB) and Rh factor (positive or negative). The sample space consists of eight possible outcomes, as each blood type can be paired with either Rh-positive or Rh-negative.

Tree diagrams are useful for visualizing all possible outcomes in a sample space.

Simple Events

A simple event 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).

The Fundamental Counting Principle

The Fundamental Counting Principle states that if one event can occur in m ways and another in n ways, the number of ways both can occur in sequence is m × n. This principle extends to any number of sequential events.

• Example: If you have 3 manufacturers, 2 car sizes, and 4 colors, the total number of combinations is .

• Example: For a four-digit access code (digits 0–9):

  • Each digit used only once: possible codes.

  • Digits can be repeated: possible codes.

  • First digit cannot be 0 or 1: possible codes.

Types of Probability

Probability can be classified into three main types:

• Classical (Theoretical) Probability: Each outcome in the sample space is equally likely. Formula: Example: Rolling a 3 on a six-sided die: .

• Empirical (Statistical) Probability: Based on observed data from experiments. Formula: Example: If 560 out of 1502 adults read only print books, .

• Subjective Probability: Based on intuition, educated guesses, or estimates. Example: A doctor estimates a 90% chance of recovery for a patient.

Law of Large Numbers

The Law of Large Numbers states that as an experiment is repeated many times, the empirical probability approaches the theoretical probability.

Range of Probabilities Rule

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

Complementary Events

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

• Example: If the probability of a user being 25–34 years old is , then the probability of not being 25–34 is .

Tree Diagrams in Probability

Tree diagrams are graphical tools used to display all possible outcomes of a probability experiment, especially when multiple events occur in sequence.

• Example: Tossing a coin and spinning a spinner. Tree diagrams help visualize and calculate probabilities for combined events, such as tossing a tail and spinning an odd number.

Using the Fundamental Counting Principle for Identification Numbers

If a college identification number consists of eight digits (0–9), and each digit can be repeated, the total number of possible identification numbers is:

Only one of these numbers matches your specific ID.

Summary Table: Types of Probability

Summary Table: Counting Principle Examples