Introduction

A random experiment is a process leading to two or more possible outcomes without knowing which outcome will occur.

The sample space is the set of all possible outcomes of a random experiment.

An event is any subset of outcomes from the sample space.

Intersection of Events

Let \(A\) and \(B\) be two events in the sample space \(S\). Their intersection, denoted by \(A \cap B\) is the set of all outcomes in \(S\) that belong to both \(A\) and \(B\).

Mutually Exclusive (Disjoint) Events

If the events \(A\) and \(B\) have no common outcomes, they are called mutually exclusive (or disjoint), and their intersection \(A \cap B\) is equal to the empty set indicating that \(A \cap B\) cannot occur.

Union of Events

Let \(A\) and \(B\) be two events in the sample space \(S\). Their union, denoted by \(A \cup B\) is the set of all outcomes in \(S\) that belong to at least one of these two events. Hence, \(A \cup B\) is made up the events in \(A\) or \(B\) or both.

Complement of an Event

Let \(A\) be an event in the sample space \(S\). The set of outcomes that belong to \(S\) but not to \(A\) is called the complement of \(A\) and is denoted by \(\bar A\) or \(A^c\).

Example 1:

Chance experiment = roll a die. Define the following events: \(A = \{\text{roll an odd number}\}\) and \(B = \{\text{roll a 1 or a 2}\}\)

Find the following events:

  • \(A^c\)
  • \(A \cap B\)
  • \(A \cup B\)

\(A = \{1,3,5\}\) and \(B = \{1, 2\}\)

  • \(A^c\) = all events not in event \(A = \{2, 4, 6\}\)
  • \(A \cap B\) = all events in both \(A\) and \(B\) = {1}$
  • \(A \cup B\) = all events in \(A\) or in \(B\) or in both \(A\) and \(B\) = \(\{1, 2, 3, 5\}\)