Lecture - Probability MT22, I

An experiment is a set $\Omega$ of possible outcomes, e.g. a specific outcome $\omega \in \Omega$.
$\Omega$ is the sample space.
An event is a subset of $\Omega$, and probabilities can be assiged to events. E.g. flipping tails might be ${T}$.
If $\omega$ is the observed outcome of an experiment, then $A$ ‘occurs’ if $\omega \in A$.
$A^C$ occurs if A does not occur.
$A \backslash B$, A occurs and B does not.
$\mathbb{P}(A) = \frac{ \vert A \vert }{ \vert \Omega \vert }$ for a finite sample space.
Counting is important to probability for finite sample spaces because you need to work out the number of outcomes relevant.
Combinatorics is the “theory of counting”.
For $X _ 1 X _ 1 X _ 1 X _ 2 X _ 2 \ldots X _ K …$ here each $X _ i$ occurs $m _ i$ times, then the number of ways of arranging them is given by $\frac{n!}{m _ 1!m _ 2!m _ 3!\ldots m _ k!}$ where $n$ is the total number of elements.
The way to think about this is by considering what it would be like if each $X _ i$ were unique and how they fall into $m _ 1$ groups.
For the case where $k = 2$, this is just the formula for the binomail coefficient. In general, it extends into the idea of a “multinomal” coefficient.
${}^n C _ m$ is the same as the number of ways of choosing a team of size $m$ from a squad of size $n$.
Consider arranging $m$ ticks and $n-m$ crosses against each person in the squad.