Math 2.0

Probability theory and stochastic processes Proposition

Addition rule for two events

Builds on Probability spaceEventInclusion–exclusion for two sets

Statement

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space and let A,B∈ℱA, B \in \mathcal{F} be events. Then P(A∪B)+P(A∩B)=P(A)+P(B).P(A \cup B) + P(A \cap B) = P(A) + P(B). Equivalently, P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B).

Remarks

If AA and BB are mutually exclusive, then P(A∩B)=0P(A \cap B) = 0 and P(A∪B)=P(A)+P(B)P(A \cup B) = P(A) + P(B). The rule is the probabilistic form of inclusion–exclusion for two sets. The inclusion–exclusion principle gives the version for nn events. A consequence is P(A∪B)≤P(A)+P(B)P(A \cup B) \le P(A) + P(B), the case n=2n = 2 of Boole's inequality.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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Cardinality of a finite set, definition, no evidence levelCardinality of a fini…definitionConstruction 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 numbersobjectRule of sum, statement, no evidence levelRule of sumstatementSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionInclusion–exclusion for two sets, statement, no evidence levelInclusion–exclusion f…statementUnion, 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 spacedefinitionAddition rule for two events, statement, no evidence levelAddition rule for two…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. Cardinality of a finite set definition
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Rule of sum statement
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Inclusion–exclusion for two sets statement
  16. Union definition
  17. Sample space definition
  18. The rationals form a field statement
  19. Universal quantifier definition
  20. Subset definition
  21. The rationals form an ordered field statement
  22. Power set definition
  23. Universal set definition
  24. Order on the reals definition
  25. Complement definition
  26. Interval definition
  27. Sigma-algebra definition
  28. Event definition
  29. Probability axioms concept
  30. Probability measure definition
  31. Probability space definition
  32. Addition rule for two events statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_axioms, Elementary consequences
About this page
Tag0BZQ6ZX584
URIm2://0BZQ6ZX584
Aliasesinclusion-exclusion-for-two-events
Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0BZQ6ZX584.
UsesProbability space, Event, Inclusion–exclusion for two sets

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