Axioms of probability

The axioms of probability are the minimum conditions that must be verified for a function defined on a set of events to consistently determine their probabilities. They were formulated by Andrei Kolmogorov in 1933.

Kolmogorov's Axioms

Sea Ω Ω {displaystyle Omega } a sample space and A{displaystyle {mathcal {A}}} a σ-algebra of subsets of Ω Ω {displaystyle Omega }.

To the Terna (Ω Ω ,A,P){displaystyle(Omega{mathcal {A}},P)} is called the space of probability, that is, a space of events (sampling space) in which possible events have been defined to consider (the σ-algebra of events) and the probability of each event (the probability function).

Properties that follow from the axioms

Other probability propositions are immediately deduced from the previous axioms:

1. P(∅ ∅ )=0{displaystyle P(emptyset)=0} (where the empty set ∅ ∅ {displaystyle emptyset } probably represents the impossible event.
2. For any events, P(S)≤ ≤ 1{displaystyle P(S)leq 1}.
3. P(Ω Ω S)=1− − P(S){displaystyle P(Omega setminus S)=1-P(S)}.
4. Yeah. A B{displaystyle Asubseq B}, then P(A)≤ ≤ P(B){displaystyle P(A)leq P(B)}.
5. P(A B)=P(A)− − P(A B){displaystyle P(Asetminus B)=P(A)-P(Acap B)}
6. P(A B)=P(A)+P(B)− − P(A B){displaystyle P(Acup B)=P(A)+P(B)-P(Acap B)}.
7. <math alttext="{displaystyle Pleft(bigcup _{i=1}^{n}{S_{i}}right)=sum _{i=1}^{n}{P(S_{i})}-sum _{i_{1}<i_{2}}^{n}{P(S_{i_{1}}cap S_{i_{2}})}+sum _{i_{1}<i_{2}P( i=1nSi)=␡ ␡ i=1nP(Si)− − ␡ ␡ i1.i2nP(Si1 Si2)+␡ ␡ i1.i2.i3nP(Si1 Si2 Si3)− − ...+(− − 1)nP( i=1nSi)#### #########################################################################################<img alt="{displaystyle Pleft(bigcup _{i=1}^{n}{S_{i}}right)=sum _{i=1}^{n}{P(S_{i})}-sum _{i_{1}<i_{2}}^{n}{P(S_{i_{1}}cap S_{i_{2}})}+sum _{i_{1}<i_{2}.
8. P( i=1nSi)≤ ≤ ␡ ␡ i=1nP(Si){displaystyle Pleft(bigcup _{i=1}^{n}{S_{i}}}}right)leq sum _{i=1}^{n}{P(S_{i})}}}}}.
9. P( i=1nSi)≥ ≥ 1− − ␡ ␡ i=1nP(Ω Ω Si){displaystyle Pleft(bigcap _{i=1}^{n}{S_{i}}}right)geq 1-sum _{i=1}^{n}{P(Omega setminus S_{i})}}}}}

Examples

We take as a sample space possible results by throwing a given Ω Ω ={1,2,3,4,5,6!{displaystyle Omega =left{1,2,3,4,5,6right}}. We'll take as σ-algebra P(Ω Ω ){displaystyle {mathcal {P}(Omega)} and as a probability function P(S)=# # S6Русский Русский S한 한 P(Ω Ω ){displaystyle P(S)={frac {#S}{6}}}quad forall Sin {mathcal {P}(Omega)}}Where # # S{displaystyle #S} represents the number of elements of the set S{displaystyle S}.

It is easy to check that this function verifies the three Kolmogorov axioms:

A1) P(S)=# # S6≥ ≥ 0{displaystyle P(S)={frac {#S}{6}}geq 0}, since it is the quotient of two positive numbers.

A2) P(Ω Ω )=P({1,2,3,4,5,6!)=# # {1,2,3,4,5,6!6=66=1{displaystyle P(Omega)=Pleft(left{1,2,3,4,5,6right}right)={frac {#left{1,2,3,4,5,6right}}}{6}}}{frac {6}}}=1}

A3) If we have S1,S2,...,Sn한 한 P(Ω Ω ){displaystyle S_{1},S_{2},...,S_{n}in {mathcal {P}}(Omega)} such that Si Sj=∅ ∅ Русский Русский iI was. I was. j{displaystyle S_{i}cap S_{j}=emptyset quad forall ineq j}, then:

# # ( i=1nSi)=␡ ␡ i=1n# # Si P( i=1nSi)=␡ ␡ i=1nP(Si){displaystyle #left(bigcup _{i=1}{n}{s_{i}}}}right)=sum _{i=1}{n}{n}{#s_{i}}{longrightarrow Pbigcup _{i=1}{i}{i}{i{i}{i}{i}{i.

Therefore, P{displaystyle P} is a probability function over (Ω Ω ,P(Ω Ω )){displaystyle(Omega{mathcal {P}}(Omega)}.