On extreme joint probabilities of~$k$ events chosen from~$n$ events

Yu. A. Zuev

Prikladnaâ diskretnaâ matematika, no. 1 (2018), pp. 5-12

Résumé

An arbitrary probability space with $n$ events is considered. All events have the same probability $p$. No restrictions on correlations between the events are imposed and the events are considered simply as arbitrary subsets of measure $p$ in the probability space. From the set of $n$ events, all $C_n^k$ subsets $X$ consisting of $k$ events are chosen, and for each such subset $X$ the probability $\mathsf P(X)$ of joint implementation of its $k$ events is considered. The subset with the minimum probability $\min_{X\colon|X|=k}\mathsf P(X)$ and the subset with the maximum probability $\max_{X\colon|X|=k}\mathsf P(X)$ are selected. In the paper, exact boundaries for both probabilities are obtained. For minimum probability: \begin{gather*} \text{if}\ kp\le k-1,\quad\text{then}\quad 0\le\min_{X\colon|X|=k}\mathsf P(X)\le p;\\ \text{if}\ kp>k-1,\quad\text{then}\quad kp-k+1\le\min_{X\colon|X|=k}\mathsf P(X)\le p. \end{gather*} For maximum probability: \begin{gather*} \text{if}\ np,\quad\text{then}\quad 0\le\max_{X\colon|X|=k}\mathsf P(X)\le p;\\ \text{if}\ k-1\le np,\quad\text{then}\quad\frac{np-\lfloor np\rfloor}{C_n^k}\le\max_{X\colon|X|=k}\mathsf P(X)\le p;\\ \text{if}\ k\le np,\quad\text{then}\quad\frac{(\lfloor np\rfloor+1-np)C_{\lfloor np\rfloor}^k +(np-\lfloor np\rfloor) C_{\lfloor np\rfloor+1}^k}{C_n^k}\le\max_{X\colon|X|=k}\mathsf P(X)\le p. \end{gather*}

Détail

Keywords: event, probability, linear programming, optimum base.