A. 583. There are given n random events (n3) such that each of them has probability , the joint probability of each two is and the joint probability of each three is . (a) Prove that the probability that none of the events occurs is at most . (b) Show that there are infinitely many numbers n, for which the events can be specified in such a way that the probability that none of the events occurs is precisely .
Based on problem 3 of the Kürschák competition in 2012
Deadline expired on 11 March 2013.
Solution. (a) Denote by N the number of the occuring events, and let p be the probability that N=0. Denote by E(f(N)|N>0) the conditional expected value of f(N), assuming N>0. Then we have
By substituting x=N into and taking conditional expected values we get
(b) If n=2k the equiality can be achieved. Suppose that
-- with probability none of the events occurs.
-- with probability all of the events occurs.
-- with probability , precisely half of the events occur, and the k-sets of events have the same probability. Then