Math 2.0

Probability theory and stochastic processes Counterexample

Pairwise independent events that are not mutually independent

Builds on Pairwise independenceMutual independence of events

Counterexample

Toss a fair coin two times, so Ω={HH,HT,TH,TT}\Omega = \{HH, HT, TH, TT\} with probability 1/41/4 for each outcome. Let AA be "the first toss is heads", BB be "the second toss is heads", and CC be "the two tosses are equal". Then A={HH,HT}A = \{HH, HT\}, B={HH,TH}B = \{HH, TH\} and C={HH,TT}C = \{HH, TT\}, so P(A)=P(B)=P(C)=1/2P(A) = P(B) = P(C) = 1/2. Each intersection of two of them is {HH}\{HH\}, with probability 1/4=12⋅121/4 = \frac12 \cdot \frac12. So AA, BB and CC are pairwise independent. But P(A∩B∩C)=P({HH})=1/4P(A \cap B \cap C) = P(\{HH\}) = 1/4, and P(A) P(B) P(C)=1/8P(A) \, P(B) \, P(C) = 1/8. So AA, BB and CC are not mutually independent.

Remarks

Here any two of the events determine the third: if AA and BB both occur, then CC occurs. The same idea with nn fair bits and their sums modulo 22 gives large pairwise independent families. Such families are useful in computer science.

Reading path

To understand this page from the ground up, read these 33 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 spacedefinitionConditional probability, definition, no evidence levelConditional probabili…definitionIndependence of two events, definition, no evidence levelIndependence of two e…definitionMutual independence of events, definition, no evidence levelMutual independence o…definitionPairwise independence, definition, no evidence levelPairwise independencedefinitionPairwise independent events that are not mutually independent, counterexample, no evidence levelPairwise independent …counterexample
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. Conditional probability definition
  30. Independence of two events definition
  31. Mutual independence of events definition
  32. Pairwise independence definition
  33. Pairwise independent events that are not mutually independent counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Pairwise_independence, Example
About this page
Tag0AR4B3K5QD
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Kindcounterexample
WritingW0 stub
MSC60A05
UsesPairwise independence, Mutual independence of events

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