Let $(B_i)$ be a collection of i.i.d. random variables taking values 0 or 1.

Suppose $0 < x^- < x^+ < 1$. Consider two different "success probabilities" for the i.i.d. collection $B_i$: under measure $P^-$, each $B_i$ takes value 1 with probability $x^-$, while under measure $P^+$, each $B_i$ takes value 1 with probability $x^+$.

Consider increasing events which are functions of the collection $(B_i)$.

I'm looking for a result of the following form: given $u^-\in(0,1)$, there exists $u^+>u^-$ such that if $A$ is an increasing event with $P^-(A)\geq u^-$, then $P^+(A)\geq u^+$.

Perhaps I'm missing something simple.... There are easy arguments, for example, when $x^+\geq (x^-)^{1/2}$. (Then one can consider two independent samples of $(B_i)$ from $P^+$, and take the min of the two samples; this dominates a sample from $P^-$. From this the result follows with $u^+=(u^-)^{1/2}$.)