Math 2.0

Probability theory and stochastic processes Proof

Proof of the law of total probability for an event and its complement

Builds on Probability spaceLaw of total probabilityProbability of the complementComplement

Proof

  1. 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 be the complement of BB.
  2. By the complement rule, P(Bc)=1−P(B)P(B^c) = 1 - P(B). Since P(B)<1P(B) < 1, we get P(Bc)>0P(B^c) > 0.
  3. The events B1=BB_1 = B and B2=BcB_2 = B^c are disjoint, and B1∪B2=ΩB_1 \cup B_2 = \Omega. Both have positive probability, by the hypothesis and step 2.
  4. The law of total probability for the family B1,B2B_1, B_2 gives P(A)=P(A∣B) P(B)+P(A∣Bc) P(Bc)P(A) = P(A \mid B) \, P(B) + P(A \mid B^c) \, P(B^c).
  5. Substituting step 2 gives 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)).

Idea

An event and its complement form the simplest partition of the sample space. The condition 0<P(B)<10 < P(B) < 1 makes both pieces have positive probability, so both conditional probabilities are defined.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof of the probability of the complement, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the law of total probability, Proof that the reals are Archimedean from the completeness axiom.

  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. Real numbers object
  9. Set bounded above definition
  10. Set definition
  11. Ring of integers statement
  12. Archimedean ordered field definition
  13. Conjunction definition
  14. Disjunction definition
  15. Implication definition
  16. Predicate definition
  17. The reals form a field statement
  18. Supremum definition
  19. Empty set object
  20. Family of sets definition
  21. Ordered pair definition
  22. Intersection definition
  23. Union definition
  24. Sample space definition
  25. The rationals form a field statement
  26. Existential quantifier definition
  27. Universal quantifier definition
  28. Completeness axiom statement
  29. Disjoint sets definition
  30. Cartesian product definition
  31. Subset definition
  32. Union of a family of sets definition
  33. The rationals form an ordered field statement
  34. Universal set definition
  35. Binary relation definition
  36. Partition of a set definition
  37. Order on the reals definition
  38. Power set definition
  39. Antisymmetric relation definition
  40. Complement definition
  41. Reflexive relation definition
  42. Transitive relation definition
  43. Interval definition
  44. Sigma-algebra definition
  45. Preorder definition
  46. Partial order definition
  47. Event definition
  48. Total order definition
  49. Probability axioms concept
  50. Probability measure definition
  51. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  52. Probability space definition
  53. The reals form an ordered field statement
  54. Conditional probability definition
  55. Proof of the approximation property of the supremum proof
  56. Approximation property of the supremum statement
  57. Proof that the reals are Archimedean from the completeness axiom proof
  58. The reals are Archimedean statement
  59. Proof of the probability of the complement proof
  60. Probability of the complement statement
  61. Proof of the law of total probability proof
  62. Law of total probability statement
  63. Proof of the law of total probability for an event and its complement proof

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_total_probability, Statement
About this page
TagEQP959VRYR
URIm2://EQP959VRYR
Kindproof
WritingW0 stub
MSC60A05
ProvesLaw of total probability for an event and its complement
UsesProbability space, Law of total probability, Probability of the complement, Complement
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.