Probability Laws

STA 211 · Lecture 03

Introduction

A probability model is a triple $\left( \Omega\mathcal{,F,}P \right)$ consisting of a sample space, a collection of events and a probability measure. The axioms are few, but they generate the principal laws used throughout probability and statistics.

Learning Objectives

You should be able to:

• define a sample space, event and probability measure;
• state and apply the probability axioms;
• use complement, addition and inclusion-exclusion laws;
• apply De Morgan’s laws; and
• distinguish mutually exclusive and exhaustive events.

Pre-Test

1. What is the sample space for two tosses of a coin?
2. Can two events overlap and still be distinct?

Views of Probability

Three common interpretations are useful:

• Classical: based on equally likely outcomes.
• Relative-frequency: probability is the long-run proportion of occurrence in repeated trials.
• Subjective or Bayesian: probability represents a coherent degree of belief, updated when evidence becomes available.

The mathematical laws are the same once probabilities have been assigned.

Axioms of Probability

For every event $A\mathcal{\in F}$:

1. Non-negativity: $P(A) \geq 0$.
2. Normalisation: $P(\Omega) = 1$.
3. Countable additivity: if $A_{1},A_{2},\ldots$ are pairwise disjoint, then

$$P\left( \underset{i = 1}{\bigcup^{\infty}}A_{i} \right) = \sum_{i = 1}^{\infty}P\left( A_{i} \right).$$

Immediate Consequences

The impossible event has probability zero:

$$P(\varnothing) = 0.$$

The complement rule is

$$P\left( A^{c} \right) = 1 - P(A).$$

If $A \subseteq B$, then

$$P(A) \leq P(B).$$

For any events $A$ and $B$,

$$P(A\backslash B) = P(A) - P(A \cap B).$$

Addition Law and Inclusion-Exclusion

For two events,

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

The subtraction is required because outcomes in $A \cap B$ are included twice in $P(A) + P(B)$.

Two events and their intersection

Two events and their intersection

For three events,

$$\begin{matrix} P(A \cup B \cup C) = & P(A) + P(B) + P(C) \\ & - P(A \cap B) - P(A \cap C) - P(B \cap C) \\ & + P(A \cap B \cap C). \end{matrix}$$

Example 3.1: Survey responses

In a survey, 60% use mobile banking, 45% use internet banking and 30% use both. The probability that a randomly selected respondent uses at least one service is

$$0.60 + 0.45 - 0.30 = 0.75.$$

The probability of using neither service is

$$1 - 0.75 = 0.25.$$

De Morgan’s Laws

The complements of unions and intersections satisfy

$$(A \cup B)^{c} = A^{c} \cap B^{c},$$

$$(A \cap B)^{c} = A^{c} \cup B^{c}.$$

Thus, “not at least one” means “none”, and “not both” means “at least one fails”.

Example 3.2: At least one success

If independent trials have success probability $p_{i}$ on trial $i$, the probability of at least one success is often easiest to calculate by the complement:

$$P\left( \text{at least one success} \right) = 1 - \prod_{i = 1}^{n}\left( 1 - p_{i} \right).$$

For four fair die rolls, the probability of at least one six is

$$1 - \left( \frac{5}{6} \right)^{4} \approx 0.5177.$$

Mutually Exclusive and Exhaustive Events

Events are mutually exclusive if

$$A \cap B = \varnothing.$$

A collection $A_{1},\ldots,A_{k}$ is exhaustive if

$$\underset{i = 1}{\bigcup^{k}}A_{i} = \Omega.$$

A mutually exclusive and exhaustive collection forms a partition of the sample space.

Common Errors

• Adding $P(A)$ and $P(B)$ without subtracting the overlap.
• Confusing “mutually exclusive” with “independent”.
• Assuming outcomes are equally likely without justification.
• Treating $P(A \cap B)$ as $P(A)P(B)$ before independence has been established.

Post-Test

1. If $P(A) = 0.55$, $P(B) = 0.40$ and $P(A \cap B) = 0.20$, find $P(A \cup B)$.
2. If $P(A \cup B) = 0.80$, $P(A) = 0.60$ and $P(B) = 0.45$, find $P(A \cap B)$.
3. State the three-event inclusion-exclusion formula.
4. A fair coin is tossed five times. Find the probability of at least one head.
5. If $A \subseteq B$, explain why $P(A) \leq P(B)$.

Selected answers: 1. $0.75$; 2. $0.25$; 4. $1 - (1/2)^{5} = 31/32$.

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