Lecture 1

A trial, experiment, or random experiment is a situation in which an outcome will occur, but we cannot be sure beforehand what will happen.

Examples include throwing a die or observing when a radioactive isotope will next decay.

An outcome is the result of an experiment. It is often denoted by $u$.

The sample space is the set of all possible outcomes. It is often denoted by $\mathrm{\Omega }$.

An event $A$ is a subset of the sample space $\mathrm{\Omega }$. The event $A$ occurs if the outcome belongs to $A$.

When rolling a die, let

Here, $A$ is the event of rolling an even number, and $B$ is the event of rolling a number greater than $3$.

The complement of $A$, denoted by $A^c$, consists of all outcomes in $\mathrm{\Omega }$ that are not in $A$. Thus, $A^c$ occurs if and only if $A$ does not occur.

In the example above,

the odd numbers.

The intersection of $A$ and $B$, denoted by $A\cap B$, consists of all outcomes that belong to both $A$ and $B$. Thus, $A\cap B$ occurs if both $A$ and $B$ occur.

The union of $A$ and $B$, denoted by $A\cup B$, consists of all outcomes that belong to $A$, to $B$, or to both. Thus, $A\cup B$ occurs if at least one of $A$ and $B$ occurs.

The set difference $A\setminus B$ consists of all outcomes that belong to $A$ but not to $B$. Thus, $A\setminus B$ occurs if and only if $A$ occurs and $B$ does not occur. Equivalently,

Given two events $A$ and $B$, we write $A\subseteq B$ if every outcome in $A$ is also in $B$. In other words, $A$ entails $B$. We then say that $A$ is a subevent, or subset, of $B$.

Two events $A$ and $B$ are mutually exclusive, or disjoint, if they have no outcomes in common, meaning

In particular, $A$ and $A^c$ are always disjoint.

Each event $A$ in a sample space $\mathrm{\Omega }$ is assigned a number $P(A)$, called the probability of $A$. The function $P$ takes values in $[\mathrm{0,1}]$ and satisfies the probability axioms.

1. Non-negativity: for every event $A$,

   $$0\le P(A)\le 1.$$

2. Normalisation:

   $$P(\mathrm{\Omega })=1.$$

3. Additivity for disjoint events: if $A\cap B=\varnothing$, then

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

The third statement is the two-event form used here. In general, Kolmogorov’s third axiom requires countable additivity: for pairwise disjoint events $A_1,A_2,\dots$,

1. Complement rule:

   $$P(A^c)=1-P(A).$$

2. Impossible event:

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

3. Addition rule:

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

$P$ is called the uniform probability measure on $\mathrm{\Omega }$ if every outcome is equally likely:

Let $\mathrm{\Omega }=\{u_1,\dots ,u_n\}$ be a finite, nonempty sample space, and let $P$ be the uniform probability measure on $\mathrm{\Omega }$. If $A\subseteq \mathrm{\Omega }$, then