Axioms of Probability

Definition: The set of all outcomes of an experiment is known as the sample space, denoted by $S$ . Any subset $E$ of $S$ is known as an event.

Observation: The union of two events $E \cup F$ happens when either event occurs. The intersection of two events $E \cap F$ (we’ll also denote with $EF$ ) when both events occur. We can also look at unions and intersections of multiple events, like $n = 1 ⋃ \infty E_{n}$ . If $E \subset F$ then $E$ is a subset of $F$ and $F$ is a superset of $E$ .

Definition: If $EF = \emptyset$ then we call the events mutually exclusive.
Definition: We also define the complement $E^{c} = S - E$

Observation

Commutative laws: $E \cup F = F \cup E$ and $EF = FE$
Associative laws: $(E \cup F) \cup G = E \cup (F \cup G)$ and $(EF) G = E (FG)$
Distributive laws: $(E \cup F) G = EG \cup FG$ and $EF \cup G = (E \cup G) (F \cup G)$

DeMorgan’s Laws: $(i = 1 ⋃ n E_{i})^{c} = i = 1 ⋂ n E_{i}^{c}$ and $(i = 1 ⋂ n E_{i})^{c} = i = 1 ⋃ n E_{i}^{c}$
Or in less fuckery (credit to Alon) notation, $(E \cup F)^{c} = E^{c} F^{c}$ and $(EF)^{c} = E^{c} \cup F^{c}$

Definition: We can define probability in terms of relative frequency, such that $P (E) = n \to \infty lim \frac{n ( E )}{n}$

Observation: This frequentist approach has confusing things about it which I will not write here but you can philosophize about. The modern approach is to build up some simple axioms of probability and then prove the existence of a limiting frequency from that.

Axiom 1: $0 \leq P (E) \leq 1$
Axiom 2: $P (S) = 1$
Axiom 3: For $EF = \emptyset$ , $P (E \cup F) = P (E) + P (F)$

Observation: $P (\emptyset) = 0$
Note that this does not account for the eagle probability (credit to Katja)

Observation: When the sample space is an uncountably infinite set, $P (E)$ is only defined for a class of events called measurable

Observation: $P (E \cup F) = P (E) + P (F) - P (EF)$

The inclusion-exclusion identity states that $P (\cup_{i = 1}^{n} E_{i}) = r = 1 \sum n (- 1)^{r + 1} i_{1} < \dots < i_{r} \sum P (E_{i_{1}} \dots E_{i_{r}})$

We can also derive bounds from these equations,
$P (\cup_{i = 1}^{n} E_{i}) \leq i = 1 \sum n P (E_{i})$
$P (\cup_{i = 1}^{n} E_{i}) \geq i = 1 \sum n P (E_{i}) - j < i \sum P (E_{i} E_{j})$
$P (\cup_{i = 1}^{n} E_{i}) \leq \dots$

Binyamin's Notes

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