Discrete Probability Space

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


Probabilities can be ascribed to points of Ω by a function \textstyle p : Ω RightArrow [0,1] such that \sumIn Ω} p(ω) = 1 . All subsets of Ω can be treated as events (thus, \textstyle \mathcal F = 2Ω 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 = 2Ω 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.