Probability

Probability is the branch of mathematics which studies the possible outcomes of given events together with their relative likelihoods and distributions. In common usage, the word ``probability'' is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a Percentage between 0 and 100%. The analysis of events governed by probability is called Statistics.

There are several competing interpretations of the actual ``meaning'' of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution.

A properly normalized function which assigns a probability ``density'' to each possible outcome within some interval is called a Probability Function, and its cumulative value (integral for a continuous distribution or sum for a discrete distribution) is called a Distribution Function.

Probabilities are defined to obey certain assumptions, called the Probability Axioms. Let a Sample Space contain the Union ( $\cup$ ) of all possible events , so

$\begin{displaymath} S\equiv \left({\,\bigcup_{i=1}^N E_i}\right), \end{displaymath}$

(1)

and let

and

denote subsets of

. Further, let $F'=\hbox{not-}F$ be the complement of

, so that

$\begin{displaymath} F\cup F'=S. \end{displaymath}$

(2)

Then the set

can be written as

$\begin{displaymath} E = E\cap S = E\cap(F\cup F') = (E\cap F)\cup (E\cap F'), \end{displaymath}$

(3)

where $\cap$ denotes the intersection. Then

$\displaystyle P(E)$	$\textstyle =$	$\displaystyle P(E\cap F)+P(E\cap F')-P[(E\cap F)\cap (E\cap F')]$
	$\textstyle =$	$\displaystyle P(E\cap F)+P(E\cap F')-P[(F\cap F')\cap (E\cap E)]$
	$\textstyle =$	$\displaystyle P(E\cap F)+P(E\cap F')-P(\emptyset \cap E)$
	$\textstyle =$	$\displaystyle P(E\cap F)+P(E\cap F')-P(\emptyset)$
	$\textstyle =$	$\displaystyle P(E\cap F)+P(E\cap F'),$	(4)

where $\emptyset$ is the Empty Set.

Let $P(E\vert F)$ denote the Conditional Probability of given that has already occurred, then

$\displaystyle P(E)$	$\textstyle =$	$\displaystyle P(E\vert F)P(F)+P(E\vert F')P(F')$	(5)
	$\textstyle =$	$\displaystyle P(E\vert F)P(F)+P(E\vert F')[1-P(F)]$	(6)
$\displaystyle P(A\cap B)$	$\textstyle =$	$\displaystyle P(A)P(B\vert A)$	(7)
	$\textstyle =$	$\displaystyle P(B)P(A\vert B)$	(8)
$\displaystyle P(A'\cap B)$	$\textstyle =$	$\displaystyle P(A')P(B\vert A')$	(9)
$\displaystyle P(E\vert F)$	$\textstyle =$	$\displaystyle {P(E\cap F)\over P(F)}.$	(10)

A very important result states that

$\begin{displaymath} P(E\cup F)=P(E)+P(F)-P(E\cap F), \end{displaymath}$

(11)

which can be generalized to

$P\left({\,\bigcup_{i=1}^n A_i\,}\right)= \sum_i P(A_i)-\setbox0=\hbox{$\scripts... ...kern\wd4\else\kern\dimen0\fi\fi \mathop{{\sum}'}_{\kern-\wd4 ij} P(A_i\cup A_j)$
$+\setbox0=\hbox{$\scriptstyle{ijk}$}\setbox2=\hbox{$\displaystyle{\sum}$}\setb... ... P(A_i\cap A_j\cap A_k)-\ldots+(-1)^{n-1}P\left({\,\bigcap_{i=1}^n A_i}\right).$
	(12)