Editing Probability theory (section)

===Discrete probability distributions===
{{Main|Discrete probability distribution}}

[[File:NYW-DK-Poisson(5).svg|thumb|300px|The [[Poisson distribution]], a discrete probability distribution]]

{{em|Discrete probability theory}} deals with events that occur in [[countable]] sample spaces.

Examples: Throwing [[dice]], experiments with [[deck of cards|decks of cards]], [[random walk]], and tossing [[coin]]s.

{{em|Classical definition}}:
Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see [[Classical definition of probability]].

For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by <math>\tfrac{3}{6}=\tfrac{1}{2}</math>, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.

{{em|Modern definition}}:
The modern definition starts with a [[Countable set|finite or countable set]] called the [[sample space]], which relates to the set of all ''possible outcomes'' in classical sense, denoted by <math>\Omega</math>. It is then assumed that for each element <math>x \in \Omega\,</math>, an intrinsic "probability" value <math>f(x)\,</math> is attached, which satisfies the following properties:
# <math>f(x)\in[0,1]\mbox{ for all }x\in \Omega\,;</math>
# <math>\sum_{x\in \Omega} f(x) = 1\,.</math>

That is, the probability function ''f''(''x'') lies between zero and one for every value of ''x'' in the sample space ''Ω'', and the sum of ''f''(''x'') over all values ''x'' in the sample space ''Ω'' is equal to 1. An {{em|[[Event (probability theory)|event]]}} is defined as any [[subset]] <math>E\,</math> of the sample space <math>\Omega\,</math>. The {{em|probability}} of the event <math>E\,</math> is defined as
:<math>P(E)=\sum_{x\in E} f(x)\,.</math>

So, the probability of the entire sample space is 1, and the probability of the null event is 0.

The function <math>f(x)\,</math> mapping a point in the sample space to the "probability" value is called a {{em|probability mass function}} abbreviated as {{em|pmf}}.