On the duration of the problem of the points

Sheldon M. Ross, Mehrdad Shahshahani, Gideon Weiss

Research output: Contribution to journalArticlepeer-review


We consider an r-player version of the famous problem of the points, which was the stimulus for the correspondence between Pascal and Fermat in the 17th century. At each play of a game, exactly one of the players wins a point, player i winning with probability pi. The game ends the first time a player has accumulated his or her required number of points—this requirement being nifor player i. A reliability application would be to suppose that a system is subject to r different types of shocks and failure occurs the first time there have been nitype i shocks for any i = 1, …, r. Our main result is to show that N, the total number of plays, is an increasing failure-rate random variable. In addition, we prove some Schur convexity results regarding P{N ≤ k) as a function of p (for ni≡ n) and as a function of n (for pi≡ 1/r).

Original languageEnglish
Pages (from-to)663-666
Number of pages4
JournalJournal of the American Statistical Association
Issue number371
StatePublished - Sep 1980
Externally publishedYes


  • Duration of play
  • Increasing failure rate
  • Problem of the points
  • Schur convex

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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