### Abstract

In this paper we deal with an optimal stopping problem whose objective is to maximize the probability of selecting k out of the last l successes, given a sequence of independent Bernoulli trials of length N, where k and l are predetermined integers satisfying 1≤k≤l<N. This problem includes some odds problems as special cases, e.g. Bruss' odds problem, Bruss and Paindaveine's problem of selecting the last l successes, and Tamaki's multiplicative odds problem for stopping at any of the last m successes. We show that an optimal stopping rule is obtained by a threshold strategy. We also present the tight lower bound and an asymptotic lower bound for the probability of a win. Interestingly, our asymptotic lower bound is attained by using a variation of the well-known secretary problem, which is a special case of the odds problem. Our approach is based on the application of Newton's inequalities and optimization technique, which gives a unified view to the previous works.

Original language | English |
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Pages (from-to) | 12-22 |

Number of pages | 11 |

Journal | Journal of Applied Probability |

Volume | 54 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 Mar 1 |

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### Keywords

- lower bound
- Newton's inequality
- odds problem
- Optimal stopping
- secretary problem

### ASJC Scopus subject areas

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

### Cite this

*Journal of Applied Probability*,

*54*(1), 12-22. https://doi.org/10.1017/jpr.2016.83