Terminology

Overview

Understanding the terminology of probability is essential for solving problems and interpreting outcomes. Below are key terms and concepts:

Probability Terminology

Experiment

An experiment is an activity or process that produces outcomes.

Outcome

An outcome is a possible result of an experiment.

Sample Space ($S$)

• The sample space is the set of all possible outcomes of an experiment.
• For a coin, $S = \{\text{Heads}, \text{Tails}\}$

Event

An event is a subset of the sample space, consisting of one or more outcomes.

Probability

• The probability of an event measures the likelihood of the event occurring, given by:

$$P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes in } S}$$

• Probability values range from 0 (impossible event) to 1 (certain event).

Mutually Exclusive Events

Events are mutually exclusive if they cannot occur simultaneously.

Independent Events

Events are independent if the outcome of one event does not affect the outcome of another.

Dependent Events

Events are dependent if the outcome of one event affects the outcome of another.

Complement of an Event

The complement of an event $E (E')$ consists of all outcomes in the sample space that are not in $E$.

Conditional Probability

The conditional probability of event $A$ given event $B$ is:

$$P(A | B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$$

Random Variable

A random variable assigns a numerical value to each outcome of an experiment.

Discrete vs. Continuous Probability

• Discrete Probability: Deals with countable outcomes (e.g., rolling a die).
• Continuous Probability: Deals with outcomes over a range (e.g., time taken to complete a task).