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# Introduction to Probability

### Definition 1

A probability space is a triple
$(\Omega, \mathcal{F}, P)$
where
$\Omega$
is a set of objects called the sample space,
$\mathcal{F}$
is a family of subsets of
$\Omega$
called events, and the probability measure
$P:\mathcal{F}\rightarrow [0,1]$
.
One key assumption we make is that
$\mathcal{F}$
is a
$\sigma$
-algebra containing
$\Omega$
, meaning that countably many complements, unions, and intersections of events in
$\mathcal{F}$
are also events in
$\mathcal{F}$
. The probability measure
$P$
must obey Kolmogorov’s Axioms.
1. 1.
$\forall A \in \mathcal{F},\ P(A) \geq 0$
2. 2.
$P(\Omega) = 1$
3. 3.
If
$A_1, A_2, \cdots\in \mathcal{F}$
and
$\forall i\ne j,\ A_i\bigcap A_j=\emptyset$
, then
$P\left(\bigcup_{i\geq 1}A_i\right) = \sum_{i\geq1}P(A_i)$
We choose
$\Omega$
and
$\mathcal{F}$
to model problems in a way that makes our calculations easy.

### Theorem 1

$P(A^c) = 1 - P(A)$

### Theorem 2 (Inclusion-Exclusion Principle)

$P\left( \bigcup_{i=1}^{n}A_i \right) = \sum_{k=1}^{n}(-1)^{k+1}\left( \sum_{1\leq i_1<\cdots

### Theorem 3 (Law of Total Probability)

If
$A_1, A_2, \cdots$
partition
$\Omega$
(i.e
$A_i$
are disjoint and
$\cup A_i = \Omega$
), then for event
$B$
,
$P(B) = \sum_iP(B\cap A_i)$

## Conditional Probability

### Definition 2

If
$B$
is an event with
$P(B)>0$
, then the conditional probability of
$A$
given
$B$
is
$P(A|B) = \frac{P(A\cap B)}{P(B)}$
Intuitively, conditional probabilty is the probability of event
$A$
given that event
$B$
has occurred. In terms of probability spaces, it is as if we have taken
$(\Omega, \mathcal{F}, P)$
and now have a probabilty measure
$P(\cdot|C)$
belonging to the space
$(\Omega, \mathcal{F}, P(\cdot|C))$
.

### Theorem 4 (Bayes Theorem)

$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

## Independence

### Definition 3

Events
$A$
and
$B$
are independent if
$P(A\cap B) = P(A)P(B)$
If
$P(B)>0$
, then
$A, B$
are independent if and only if
$P(A|B) = P(A)$
. In other words, knowing
$B$
occurred gave no extra information about
$A$
.

### Definition 4

If
$A,B,C$
with
$P(C)>0$
satisfy
$P(A\cap B|C) = P(A|C)P(B|C)$
, then
$A$
and
$B$
are conditionally independent given
$C$
.
Conditional independence is a special case of independence where
$A$
and
$B$
are not necessarily independent in the original probability space which has the measure
$P$
, but are independent in the new probability space conditioned on
$C$
with the measure
$P(\cdot|C)$
.