Mathematical and Physical Journal
for High Schools
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# Problem A. 583. (February 2013)

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

(5 pont)

Deadline expired on March 11, 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=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

### Statistics:

 3 students sent a solution. 5 points: Janzer Olivér, Nagy Bence Kristóf, Szabó 928 Attila.

Problems in Mathematics of KöMaL, February 2013