Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 214 / 1 / a

For bond percolation, the Harris-FKG inequality states that any two increasing events $A,B$ satisfyThe same inequality holds for two decreasing events, either by reversing the coordinate order or by taking complements in the two-event identity. Intersections of decreasing events are decreasing events, so induction givesWrite $a={\mathbb{P}}_p(A_1)=\cdots ={\mathbb{P}}_p(A_n)$ and $u={\mathbb{P}}_p({\bigcup }_j{A}_j)$. Then $1-u\ge (1-a{)}^n$. Taking the nonnegative $n$-th root and rearranging yields the square-root trick for positively associated events:No independence among the events is needed.