Math 2.0

Probability theory and stochastic processes Corollary

Law of total probability for an event and its complement

Builds on Law of total probabilityProbability of the complement

Statement

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space and let A,B∈ℱA, B \in \mathcal{F} with 0<P(B)<10 < P(B) < 1. Let Bc=Ω∖BB^c = \Omega \setminus B. Then P(A)=P(A∣B) P(B)+P(A∣Bc) (1−P(B)).P(A) = P(A \mid B) \, P(B) + P(A \mid B^c) \, (1 - P(B)).

Remarks

This is the law of total probability for the partition {B,Bc}\{B, B^c\}, together with P(Bc)=1−P(B)P(B^c) = 1 - P(B). The formula is the denominator of Bayes' theorem in the case of two hypotheses.

Reading path

To understand this page from the ground up, read these 40 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 integersstatementConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementUniversal set, definition, no evidence levelUniversal setdefinitionPartition of a set, definition, no evidence levelPartition of a setdefinitionPower set, definition, no evidence levelPower 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 spacedefinitionConditional probability, definition, no evidence levelConditional probabili…definitionProbability of the complement, statement, no evidence levelProbability of the co…statementLaw of total probability, statement, no evidence levelLaw of total probabil…statementLaw of total probability for an event and its complement, statement, no evidence levelLaw of total probabil…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. Conjunction definition
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Empty set object
  15. Family of sets definition
  16. Intersection definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Disjoint sets definition
  23. Subset definition
  24. Union of a family of sets definition
  25. The rationals form an ordered field statement
  26. Universal set definition
  27. Partition of a set definition
  28. Power set definition
  29. Order on the reals definition
  30. Complement definition
  31. Interval definition
  32. Sigma-algebra definition
  33. Event definition
  34. Probability axioms concept
  35. Probability measure definition
  36. Probability space definition
  37. Conditional probability definition
  38. Probability of the complement statement
  39. Law of total probability statement
  40. Law of total probability for an event and its complement statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_total_probability, Statement
About this page
Tag0W62TKMW7V
URIm2://0W62TKMW7V
Kindstatement, corollary
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0W62TKMW7V.
UsesLaw of total probability, Probability of the complement

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