Lecture - Probability, Lecture 1
Notes
- 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.