KURSCHAK 2012 Problem 3

Consider $n$ events, each of which has probability $\frac12$ . We also know that the probability of any two both happening is $\frac14$ . Prove the following. (a) The probability that none of these events happen is at most $\frac1{n+1}$ . (b) We can reach equality in (a) for infinitely many $n$ .