Probability

Overview

There are generally two schools of thought on how to go about interpreting a probability space.

The frequentist view of probability is that is represents a long-running frequency over a large number of repetitions of an experiment.
The Bayesian view of probability is that it represents a degree of belief about the event in question.

Probability Spaces

The sample space $S$ of an experiment is the set of all possible outcomes of the experiment. An event $A$ is a subset of the sample space $S$ . We say $A$ occurred if the actual outcome of the experiment is a member of $S$ .

Assume a finite sample space $S$ in which every outcome is equally likely, one can naïvely define the probability of an event $A \subseteq S$ as

P_{naïve} (A) = \frac{| A |}{| S |} = \frac{no. of outcomes favorable to A}{total number of outcomes in S}

More generally, a probability space $⟨ S, P ⟩$ is a (possibly infinite) sample space $S$ and a probability function $P$ which takes an event $A \subseteq S$ as input and returns $P (A)$ , a real number between $0$ and $1$ , as output. The function $P$ must satisfy the following three axioms:

$P (\emptyset) = 0$
$P (S) = 1$
If $A_{1}, A_{2}, \dots$ are disjoint events, then $P (⋃_{n = 1}^{\infty} A_{n}) = \sum_{n = 1}^{\infty} P (A_{n})$

Properties

Let $⟨ S, P ⟩$ be a probability space. If $A, B \subseteq S$ are events, the following properties hold:

$P (A^{C}) = 1 - P (A)$
$P (A △ B) = P (A) + P (B) - 2 P (A \cap B)$
$P (A - B) = P (A) - P (A \cap B)$
$A \subseteq B ⟹ P (A) \leq P (B)$

Odds

Let $⟨ S, P ⟩$ be a probability space and $A \subseteq S$ be an event. The odds of $A$ are defined as

odds (A) = \frac{P (A)}{P (A^{C})}

If $odds (A) = N$ , we say the odds in favor of $A$ are $N$ to $1$ . This is sometimes instead denoted as $N : 1$ . Equivalently, we may say the odds are against $A$ $1$ to $N$ .