Math 2.0

Probability theory and stochastic processes Markov processes Problem

Gambler's ruin

Builds on Random walkLaw of total probability

Problem

A gambler starts with kk units of money, where 0≤k≤N0 \le k \le N for an integer N≥1N \ge 1. In each round, the gambler wins one unit with probability p∈(0,1)p \in (0, 1) and loses one unit with probability q=1−pq = 1 - p. The rounds are independent. The game stops when the fortune reaches 00 (ruin) or NN (the target). Find the probability PkP_k that the gambler reaches NN before 00.

Remarks

The fortune is a random walk on {0,1,…,N}\{0, 1, \dots, N\} that stops at 00 and NN. By the law of total probability for the first round, Pk=pPk+1+qPk−1P_k = p P_{k+1} + q P_{k-1} for 0<k<N0 < k < N, with P0=0P_0 = 0 and PN=1P_N = 1. For a fair game, p=q=1/2p = q = 1/2, the solution is Pk=k/NP_k = k/N. For p≠qp \ne q, the solution is Pk=1−(q/p)k1−(q/p)NP_k = \dfrac{1 - (q/p)^k}{1 - (q/p)^N}. In a fair game against an opponent with unlimited money (N→∞N \to \infty), the gambler is ruined with probability 11.

History

An early form of the problem appears in a letter of Blaise Pascal to Pierre de Fermat in 1656.

Reading path

To understand this page from the ground up, read these 58 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. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Intersection definition
  19. Union definition
  20. Sample space definition
  21. The rationals form a field statement
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Disjoint sets definition
  25. Cartesian product definition
  26. Subset definition
  27. Union of a family of sets definition
  28. The rationals form an ordered field statement
  29. Uniqueness quantifier definition
  30. Universal set definition
  31. Binary relation definition
  32. Partition of a set definition
  33. Order on the reals definition
  34. Power set definition
  35. Complement definition
  36. Function definition
  37. Interval definition
  38. Preimage of a set definition
  39. Sigma-algebra definition
  40. Event definition
  41. Borel sigma-algebra definition
  42. Probability axioms concept
  43. Probability measure definition
  44. Probability distribution definition
  45. Probability space definition
  46. Conditional probability definition
  47. Random variable definition
  48. Independence of two events definition
  49. Law of a random variable definition
  50. Law of total probability statement
  51. Stochastic process concept
  52. Mutual independence of events definition
  53. Identically distributed random variables definition
  54. Independent sigma-algebras definition
  55. Independence of random variables definition
  56. Independent and identically distributed random variables definition
  57. Random walk definition
  58. Gambler's ruin problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Gambler's_ruin, Fair coin flipping
  2. Web https://en.wikipedia.org/wiki/Gambler's_ruin, Unfair coin flipping
About this page
Tag0WFMKK7DFZ
URIm2://0WFMKK7DFZ
Aliasesgamblers-ruin-problem
Kindproblem
WritingW1 stated
MSC60G50, 60G40
StatusThis copy of the site has no build data. Run m2 status 0WFMKK7DFZ.
UsesRandom walk, Law of total probability
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