KöMaL 2013 A583

There are given $n$ random events ($n \ge$3) such that each of them has probability $\frac12$, the joint probability of each two is $\frac14$ and the joint probability of each three is $\frac18$. ($a$) Prove that the probability that none of the events occurs is at most $\frac{1}{2n}$ . ($b$) 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 $\frac{1}{2n}$ .