# Discrete Probability Space

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A Discrete Probability Space is a Probability Space that can be mapped to a Finite Set.

**AKA:**Finite Probability Space.**Context:**- It can be:
- …

**Example(s):**- Ω = {H,T}, a Coin Toss Experiment.

**See:**Continuous Probability Space.

## References

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Probability_space#Discrete_case
- Discrete probability theory needs only at most countable sample spaces ω, which makes the foundations much less technical.

Probabilities can be ascribed to points of **Ω** by a function \textstyle p : **Ω** RightArrow [0,1] such that \sum_{{ω In Ω}} p(ω) = 1 .
All subsets of **Ω** can be treated as events (thus, \textstyle \mathcal F = 2^{Ω} is the power set).
The probability measure takes the simple form

- If the space concerns one flip of a fair coin, then the outcomes are heads and tails: Ω = {H,T}. The σ-algebra \mathcal F = 2
^{Ω}contains \textstyle 2^2 = 4 events, namely, \textstyle \{H\} : heads, \textstyle \{T\} : tails, \textstyle \{\} : neither heads nor tails, and {H,T} : heads or tails. So, [math]\displaystyle{ F }[/math] = { {}, {H}, {T}, {H,T}}.

There is a fifty percent chance of tossing either heads or tail: *p*(H) = *p*(T) = 0.5; thus P({H}) = P({T}) = 0.5. The chance of tossing neither is zero: P({})=0, and the chance of tossing one or the other is one: \textstyle P(\{H,T\})=1.