Math 2.0

Probability theory and stochastic processes Problem

Monty Hall problem

Builds on Conditional probabilityBayes' theorem

Problem

A game show has three closed doors. A car is behind one door, chosen uniformly at random, and a goat is behind each of the two other doors. The player chooses one door. Then the host, who knows where the car is, opens one of the two other doors and shows a goat. If both other doors hide goats, the host chooses one of them with probability 1/21/2 each. The host always opens a door and always offers a switch. The player can keep the first door or switch to the other closed door. Which choice gives the larger probability of the car?

Remarks

Under these assumptions, switching wins with probability 2/32/3 and staying wins with probability 1/31/3. The player's first door hides the car with probability 1/31/3, and the host's action does not change this. The conditional probability that switching wins, given the door that the host opens, is also 2/32/3. Bayes' theorem gives this. The answer depends on the rules of the host. If the host opens a random other door and it shows a goat by chance, both choices win with probability 1/21/2.

History

Steve Selvin posed the problem in a letter to The American Statistician in 1975. It became famous through the column of Marilyn vos Savant in Parade magazine.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

Show the path as a graph
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…definitionLaw of total probability, statement, no evidence levelLaw of total probabil…statementBayes' theorem, statement, no evidence levelBayes' theoremstatementMonty Hall problem, problem, no evidence levelMonty Hall problemproblem
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. Law of total probability statement
  39. Bayes' theorem statement
  40. Monty Hall problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Monty_Hall_problem, Standard assumptions
About this page
Tag0PBWWQD9KY
URIm2://0PBWWQD9KY
Kindproblem
WritingW1 stated
MSC60C05
StatusThis copy of the site has no build data. Run m2 status 0PBWWQD9KY.
UsesConditional probability, Bayes' theorem
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