Probability Space

A probability space is a measure space equipped with probability measure.

Description

A probability space is a triple (Ω, ℱ, P), composed of:

• a sample space Ω

• an event space ℱ, noting this is a σ algebra

• a probability measure P : ℱ -> [0,1] satisfying three conditions called the Kolmogorov axioms

A closely related concept is a random variable, which itself is just a map between a sample space and some measurable space (Ω, 𝒜): X : Ω -> 𝒜. When the sample space Ω is common to both the random variable and a probability space, it is possible to lift the probability measure P from the latter into the former. This pushforward measure is a probability distribution. The probability that the random variable X takes on the value x ∈ 𝒜 is given by P(X = x) = P({ω ∈ Ω | X(ω) = x}). The probability that the random variable takes on any value in the subset S ⊆ 𝒜 is given by P(X ∈ S) = P({ω ∈ Ω | X(ω) ∈ S}).

(It may be more appropriate to say that the random variable maps to a topological space. Practically speaking, probability is applied in two ways: the sample space is countable and discrete, or the sample space is uncountable but the random variable maps to real numbers. In either case, the Borel set of Ω is the smallest possible σ algebra that can be defined on Ω.)

In the case of a discrete random variable X, f is called a probability mass function and the final condition above is expressed as:

In the case of a continuous random variable X, f is called a probability density function and the final condition above is expressed as:

Theorems and Properties of Probability

Many important properties of the probability measure derive from the properties of sets. For example

• Prob(A^C ⋂ B^C) = Prob((A ⋃ B)^C)

• Prob(A ⋂ (B ⋃ C)) = Prob((A ⋂ B) ⋃ (A ⋂ C))

Others take advantage of the definition of the probability measure.

• Prob(Omega)) = 1 so Prob(A) = 1 - Prob(A^C)

• A ⊆ B implies Prob(A) ≤ Prob(B)

The probability of a union of disjoint sets is simply the sum of their individual probabilities.

• Prob(A ⋃ B) = Prob(A) + Prob(B) for any A,B such that A ⋂ B = Ø

More generally, the probability of a union of 2 events is

• Prob(A ⋃ B) = Prob(A) + Prob(B) - Prob(A ⋂ B)

And the probability of a union of 3 events is

• Prob(A ⋃ B ⋃ C) = Prob(A) + Prob(B) + Prob(C) - Prob(A ⋂ B) - Prob(B ⋂ C) - Prob(A ⋂ C) + Prob(A ⋂ B ⋂ C)

This generalizes to n events as

• \sum \nolimits_i Prob( A_i ) - sum \nolimits_{j<i}^{n Prob( A_i \cap A_j ) + \sum \nolimits_{k<j<i}}n Prob( A_i \cap A_j \cap A_k ) - \dots + (-1)^{{n+1} Prob( \bigcap_{m=1}}n A_m)

Several useful identities derive from Bayes' theorem, like:

• Prob(A ⋂ B) = Prob(A) Prob(B|A)

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