Convention:
N = {0,1,2,...}
N* = {1,2,3,...}

HW problems 1.11 #1-4 (in book)

The unit square, with #Omega = \infty, has probability zero for lines.

Developing a single framework for discrete {heads, tails} and picking a
point from [0,1].

Ex:

Probability of getting a prime number choosing at random from N*

Def: Let Omega be a sample space. An event space E is a non-empty
collection of subsets of Omega such that:
    if A \in E, A^c \in E
        Lemma Def: iff B \in S and B \not\in A, B \in A^c
    if A_1,A_2,...,A_n \in E, then the union is in E.

Def: An event is called *elementary* if its cardinality is 1

In general, the power set of Omega (2^Omega = \bb P(Omega)) is the
collection of all subsets of omega. (sometimes called complete/total
event space)

Trivial event space: {empty, Omega}

Take Omega = {1,..,7}.
E_1 = 2^Omega. #E_1 = 2^#Omega = 2^7.
E_2 = { {3}, {1,2,4,5,6,7}, empty, Omega}. This is also a valid event sp

In general, regardless of cardinality of Omega, we will take E = 2^Omega
because all event spaces are subsets of the power set.

N, N*, Q, Z, ...
are infinite countable sets. There is a bijection between these sets and
N.

R is an infinite uncountable set. Rigorously defining Omega = R is very
difficult. But this is outside the scope of this course.

Def: A probability measure is a set function, usually denoted by P, from
E to [0,+\infty] = R^+

(i) P(Omega) = 1.
(ii) P(Union of A_1...A_n) = \sum_i=1^n P(A_i) if A_1..A_n is pairwise
disjoint.

Def: Pairwise disjoint if A_i intersect A_j = empty for all i != j.

Function:
    P: E -> [0,+\infty)
        For all A in E, there exists a unique P(A)

Upside down A = for all
Backwards E = there exists
Backwards E ! = there exists a unique

Review set operations, i.e. union, intersection, complementation,
commutativity, distributivity, associativity

For Omega finite or infinite,

\sum_\Omega P({\omega_i}) = P(\Omega) = 1.
P({\omega_i}) \geq 0.

Uniform probabliity:
    P({\omega_i}) = p_i = 1/#\Omega
    Each element is equally likely.
    If A \subset \Omega, P(A) = #A/#Omega