Set Theory and Probability Revision Notes for Leaving Cert Mathematics

Set Theory and Probability

Overview

Set theory and probability are foundational concepts in statistics that help analyse collections of items and their likelihood of occurrence in different situations.

Set Theory Basics

A set is a collection of distinct objects, which could be numbers, people, or events.

Key terms in set theory include:

Universal Set ( $U$ ): The set of all possible elements in a context.
Subset ( $A \subseteq B$ ): A set $A$ is a subset of $B$ if all elements of $A$ are also in $B$
Union ( $A \cup B$ ): Combines all elements from sets $A$ and $B$ .
Intersection ( $A \cap B$ ): Includes elements common to both $A$ and $B$
Complement ( $A^c$ ): The set of all elements in $U$ not in $A$
Disjoint Sets: Sets with no elements in common ( $A \cap B = \emptyset$ )

Set Theory Diagram

Probability Basics

Probability measures the likelihood of an event occurring. It is represented on a scale from 0 (impossible) to 1 (certain).

Sample Space ( $S$ ): The set of all possible outcomes.
Event: A subset of the sample space.
Probability Formula: For equally likely outcomes, the probability of an event $A$ is:

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

Rules of Probability

Addition Rule:

For events $A$ and $B$ :

P(A \cup B) = P(A) + P(B) - P(A \cap B)

If $A$ and $B$ are disjoint:

P(A \cup B) = P(A) + P(B)

Multiplication Rule:

For independent events $A$ and $B$ :

P(A \cap B) = P(A) \times P(B)

Worked Examples

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Example 1: Rolling a Die

Problem: What is the probability of rolling a 4 or a 6 with a fair six-sided die?

Solution:

Step 1: Define $A$ : rolling a 4, and $B$ : rolling a 6.

Step 2: Sample space $(S) = {1, 2, 3, 4, 5, 6}$ .

Step 3: Using the addition rule for disjoint events:

P(A \cup B) = P(A) + P(B) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}

Answer: The probability is $\frac{1}{3}$

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Example 2: Tossing Two Coins

Problem: What is the probability of getting at least one head when tossing two coins?

Solution:

Step 1: Define $A$ : at least one head.

Step 2: Sample space ( $S$ ) $= {HH, HT, TH, TT}$ .

Step 3: Complement of $A (A^c)$ $= {TT}$ .

Step 4: Calculate $P(A^c)$ :

P(A^c) = \frac{1}{4}

Step 5: Use the complement rule:

P(A) = 1 - P(A^c) = 1 - \frac{1}{4} = \frac{3}{4}

Answer: The probability is $\frac{3}{4}$ .

Summary

Set Theory Terms: Union ( $A \cup B$ ), Intersection ( $A \cap B$ ), Complement ( $A^c$ ), and Disjoint Sets.
Probability Rules:
- Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
- Multiplication Rule: $P(A \cap B) = P(A) \times P(B)$ (for independent events).
Key Formula: $P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$

Set Theory and Probability (Leaving Cert Mathematics): Revision Notes