Formal Methods II

Venn’s identity “probability = frequency” clearly fails
in small samples.
(or 1) just because the coin
was only flipped once.
von Mises moved to probabilities as limit frequencies in
large samples.

Definition of a collective:
An infinite sequence of outcomes S = (s1, ...., Sn, ...) of a repeatable experiment (where {si} is the set of all its possible outcomes), satisfying the conditions:
1. Axiom of Convergence

Let be the set of all possible outcomes of a repeat-
able experiment. Let the sequence be
composed of outcomes in that set (each for some
). Assuming satisfies von Mises’ definition for a
collective, show that the corresponding probability function
satisfies Kolmogorov’s axioms for a probability function.
That is, taking defined over all the singleton sets of out-
comes and their unions, show that it satisfies