Probabilities of competing binomial random variables

Wenbo V. Li, Vladislav V. Vysotsky

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Suppose that both you and your friend toss an unfair coin n times, for which the probability of heads is equal to α. What is the probability that you obtain at least d more heads than your friend if you make r additional tosses? We obtain asymptotic and monotonicity/convexity properties for this competing probability as a function of n, and demonstrate surprising phase transition phenomenon as the parameters d, r, and α vary. Our main tools are integral representations based on Fourier analysis.

Original languageEnglish (US)
Pages (from-to)731-744
Number of pages14
JournalJournal of Applied Probability
Issue number3
StatePublished - Sep 2012


  • Binomial random variable
  • Coin tossing
  • Competing random variables
  • Number of successes
  • Phase transition
  • Probability of winning

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty


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