1 Introduction

A sample space, $\Omega$, is the set of all possible outcomes of some experiment.
An event is a subset of the sample space.

1.1 Examples

Discrete Sample Space

Experiment: Roll a die until a ’1’ appears.
Sample space $\Omega = \{(1), (2, 1), (3, 1), \ldots, (2, 2, 1), \ldots, (2, 3, 4, 5, 6, 1, \ldots)\}$.
Event that the sum of values is 5: $\{(4, 1), (2, 2, 1)\}$
Tuples () consider the order of elements in it. Sets {} do not.

Continuous Sample Space

Experiment: Output any real number in $[0, 1)$.
Sample space $\Omega = [0, 1)$.
Example of Events: $\emptyset$, [0, 1), [0.1, 0.3], [0.1, 0.3] $\cup$ (0.5, 0.7)
For continuous sample space, we should define events in terms of (operations of) intervals.

1.2 Probability Distribution

1.2.0.1 Definition

Let $\Omega$ be the sample space and $\mathcal{E}$ be the set of all events (where an event is a subset of $\Omega$). A probability distribution/measure is a function $P: \mathcal{E} \rightarrow [0,1]$, if it satisfies the following axioms:

1. $0 \leq P(A) \leq 1, \quad \forall A \in \mathcal{E}$

2. $P(\Omega) = 1$

3. $A_1, A_2, \ldots \text{ are disjoint events in }\mathcal{E} \Rightarrow P(\cup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$

1.2.0.2 Properties

1. $P(\emptyset) = 0$

2. $A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)$

3. $P(A^C) = 1 - P(A)$

4. $A \subset B \Rightarrow P(A) \leq P(B)$

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

1.3 Independent vs Disjoint events

An independent event does not affect the probability of occurence of another independent event. They are related by $$P(A \cap B) = P(A)P(B)$$ Disjoint events can never occur at the same time. If A and B are disjoint events, then $$A \cap B = \emptyset \Rightarrow P(A \cap B) = 0$$ In general, independent events are not disjoint and disjoint events are not independent. The only exception is the trivial case, when one of the events is impossible i.e. $P(A)=0$, hence satisfying both equations above.