Math 2.0

concept

Probability axioms

Summary

The probability axioms of Kolmogorov are the three rules that a probability function PP on the events of a sample space Ω\Omega must obey. First, P(A)≥0P(A) \ge 0 for each event AA. Second, P(Ω)=1P(\Omega) = 1. Third, for each sequence A1,A2,…A_1, A_2, \dots of pairwise disjoint events, P(⋃nAn)=∑nP(An)P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n). This is countable additivity. A function that obeys these rules is a probability measure.

Remarks

Andrey Kolmogorov stated these axioms in 1933. The other rules of probability follow from the axioms. Examples are P(∅)=0P(\varnothing) = 0, P(A)≤1P(A) \le 1, the rule for the complement and the addition rule for two events.

Explanations

L1 human

The axioms say three things: a probability is never negative, something always happens, and the probabilities of events that cannot happen together add up. For a fair coin, P({H})=P({T})=1/2P(\{H\}) = P(\{T\}) = 1/2, and P({H,T})=1/2+1/2=1P(\{H, T\}) = 1/2 + 1/2 = 1.

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  • edge instance_of in: at least two examples (0 of 2)
  • edge not_instance_of in: at least one non-example (0 of 1)
  • edge specializes both: the more general notions and the special cases (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Probability_axioms, Kolmogorov axioms

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