**A. 583.** There are given *n* random events (*n*3) 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

(5 points)

**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

*E*(1|_{N>0})=1,

By substituting *x*=*N* into and taking conditional expected values we get

(b) If *n*=2*k* 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