Math 2.0

Probability theory and stochastic processes Example

Birthday problem

Builds on Classical definition of probabilityProbability of the complement

Example

Assume that each person has a birthday on one of 365365 days, and that each day is equally likely. Assume also that the birthdays of different people are independent. Let p(n)p(n) be the probability that among nn people, at least two have the same birthday. By the rule for the complement, for 1≤n≤3651 \le n \le 365, p(n)=1−365⋅364⋯(365−n+1)365n.p(n) = 1 - \frac{365 \cdot 364 \cdots (365 - n + 1)}{365^n}. Then p(22)≈0.476p(22) \approx 0.476 and p(23)≈0.507p(23) \approx 0.507. So 2323 people are enough for a probability above 1/21/2.

Remarks

The answer is surprising because there are many pairs: 2323 people form (232)=253\binom{23}{2} = 253 pairs. For nn much smaller than 365365, p(n)≈1−e−n(n−1)/730p(n) \approx 1 - e^{-n(n-1)/730}. For n=70n = 70, p(n)>0.999p(n) > 0.999. For n≥366n \ge 366, p(n)=1p(n) = 1 by the pigeonhole principle.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

  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. Ordered pair definition
  14. Union definition
  15. Sample space definition
  16. The rationals form a field statement
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. The rationals form an ordered field statement
  22. Uniqueness quantifier definition
  23. Binary relation definition
  24. Universal set definition
  25. Order on the reals definition
  26. Power set definition
  27. Complement definition
  28. Function definition
  29. Interval definition
  30. Codomain definition
  31. Injective function definition
  32. Sigma-algebra definition
  33. Surjective function definition
  34. Event definition
  35. Bijective function definition
  36. Probability axioms concept
  37. Equinumerous sets definition
  38. Probability measure definition
  39. Finite set definition
  40. Probability space definition
  41. Classical definition of probability definition
  42. Probability of the complement statement
  43. Birthday problem example

Further reading

  1. Web https://en.wikipedia.org/wiki/Birthday_problem, Calculating the probability
About this page
Tag0NMWG2PHJ0
URIm2://0NMWG2PHJ0
Aliasesbirthday-paradox
Kindexample
WritingW0 stub
MSC60C05
UsesClassical definition of probability, Probability of the complement
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