Square-root bound for increasing events
= Square-root bound for increasing events
{title2=$P(A)\ge1-\sqrt{1-P(A\cup B)}$}
For two <increasing events> of equal <probability> $a$ under a <product measure>, positive association of their complements gives $1-P(A\cup B)\ge(1-a)^2$. Hence $a\ge1-\sqrt{1-P(A\cup B)}$. For $r$ equal-probability <increasing events>, the same reasoning gives $a\ge1-(1-P(\bigcup_jA_j))^{1/r}$.