Classical Probability is Probability Measure
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Link to a page expanding on that statement -- perhaps better to say that it's the definition of the image of Pr?
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Theorem
The classical probability model is a probability measure.
Proof
We check all the Kolmogorov axioms in turn:
First Axiom
From Empty Set is Subset of All Sets and from the definitions of the event space and sample space:
- $\varnothing \subseteq \Sigma \subseteq \Omega $
From Cardinality of Empty Set and Cardinality of Subset of Finite Set:
- $0 \le \card \Sigma \le \card \Omega$
Dividing all terms by $\card \Omega$:
- $0 \le \dfrac {\card \Sigma} {\card \Omega} \le 1$
The middle term is the asserted definition of $\map \Pr {\, \cdot \,}$.
Link to a page expanding on that statement -- perhaps better to say that it's the definition of the image of Pr?
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$\Box$
Second Axiom
- $\map \Pr \Omega = \dfrac {\card \Omega} {\card \Omega} = 1$
$\Box$
Third Axiom
Follows from Cardinality is Additive Function and the corollary to the Inclusion-Exclusion Principle.
$\blacksquare$
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