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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 $E_i$, so

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

and let $E$ and $F$ denote subsets of $S$. Further, let $F'=\hbox{not-}F$ be the complement of $F$, so that
F\cup F'=S.
\end{displaymath} (2)

Then the set $E$ can be written as
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 $E$ given that $F$ 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
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...
\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).$


See also Bayes' Formula, Conditional Probability, Distribution, Distribution Function, Likelihood, Probability Axioms, Probability Function, Probability Inequality, Statistics

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© 1996-9 Eric W. Weisstein