General Definition of Probability

A probability space consists of a sample space $S$ and a probability function $P$ which takes an event $A\subset S$ as inputs and returns $P(A)$, a real number between $0$ and $1$, as output. The function $P$ must satisfy the following axioms:

1. $P(\varnothing )=0,P(S)=1$
2. If $A_1,A_2,\cdots$ are disjoint events, then

$$P\left(\bigcup _{j=1}^{\infty }{A}_j\right)=\sum _{j=1}^{\infty }P(A_j)$$

(Saying that these events are disjoint means that they are mutually exclusive: $A_i\cap A_j=\varnothing$ for $i\ne j$)

Any function $P$ (mapping events to numbers in the interval $[0,1]$) that satisfies the two axioms is considered a valid probability function. However, the axioms don't tell us how probability should be interpreted; different schools of thought exists

The frequentist view of probability is that it represents a long-run 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, so we can assign probabilities to hypotheses like "candidate A will win the election" even if it isn't possible to repeat the same election over and over again

We can prove probability has the following properties, for any events $A$ and $B$ :

1. $P(\bar{A})=1-P(A)$
2. If $A\subset B$, then $P(A)\le P(B)$
3. $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ The third property is a special case of inclusion-exclusion