On extreme joint probabilities of k events chosen from n events
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/k subsets X consisting of k events are chosen, and for each such subset X the probability P(X) of joint implementation of its k events is considered. The subset with the minimum probability min P(X) and X:|X|=fc the subset with the maximum probability max P(X) are selected. In the paper, X :|X |=fc exact boundaries for both probabilities are obtained. For minimum probability: if kp @ k - 1, then 0 @ min P(X) @ p; X:|X|=fc if kp > k - 1, then kp - k + 1 @ min P(X) @ p. X:|X|=fc For maximum probability: if np < k - 1, then 0 ^ max P(X) ^ p; X:|X|=fc if k if k ^ np, then, np - IПРI 1 ^ np < k, then -^ max P(X) ^ p; СП X:|X|=fc <sup>v 7</sup> ^ max P(X) ^ p. X :|X |=fc <sup>(</sup>M +<sup>1 - n</sup>p<sup>)C</sup> f<sub>rapJ</sub> + <sup>(n</sup>p <sup>-</sup> L<sup>n</sup>pJ<sup>)C</sup>f<sub>rapJ+</sub>i C k <sup>C</sup>n