### Abstract

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 language | English (US) |
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Pages (from-to) | 731-744 |

Number of pages | 14 |

Journal | Journal of Applied Probability |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2012 |

### Keywords

- 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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## Cite this

Li, W. V., & Vysotsky, V. V. (2012). Probabilities of competing binomial random variables.

*Journal of Applied Probability*,*49*(3), 731-744. https://doi.org/10.1239/jap/1346955330