Discrete Probability Space

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

• AKA: Finite Probability Space.
• Context:
  • It can be:
    • A Binary Probability Space.
    • A Multivalued Probability Space.
  • …
• 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

• • • P(A) = \sum_{{ω In A}} p(ω) ForAll A Subset Ω.
  • 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.

---