Math 2.0

Probability theory and stochastic processes Proposition

Monotonicity of probability

Builds on Probability spaceEventSubset

Statement

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space and let A,B∈ℱA, B \in \mathcal{F} be events with A⊆BA \subseteq B. Then P(A)≤P(B)P(A) \le P(B). More precisely, P(B∖A)=P(B)−P(A)P(B \setminus A) = P(B) - P(A).

Remarks

The proof: BB is the disjoint union of AA and B∖AB \setminus A, so P(B)=P(A)+P(B∖A)P(B) = P(A) + P(B \setminus A), and P(B∖A)≥0P(B \setminus A) \ge 0. With B=ΩB = \Omega, monotonicity gives P(A)≤1P(A) \le 1 for each event AA. If AA implies BB, then BB is at least as likely as AA. For example, "the die shows 66" is at most as likely as "the die shows an even number".

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionMonotonicity of probability, statement, no evidence levelMonotonicity of proba…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition
  28. Probability space definition
  29. Monotonicity of probability statement

Where this leads

20 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_axioms, Elementary consequences
About this page
Tag0464Z0JP2E
URIm2://0464Z0JP2E
Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0464Z0JP2E.
UsesProbability space, Event, Subset

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