This question is motivated by a problem on the harmonic measure. I also asked this on Mathoverflow, copying it here, and hope that cross-posting here is not a problem.
In each trial of a sequence of independent trials, we have a probability of success $p_i$ and a probability of failure $(1-p_i)$, and for all $i\in \mathbb{N}$, there exist uniform constants $\eta_1,\eta_2$ so that $\frac{1}{2}+\eta_1 \leq p_i \leq 1-\eta_2 $.
For $n$ many trials, and as $n\to \infty$, with the parameters $p_i$ in the above range, is it true that one can extract some events whose total probability is arbitrarily large compared to the corresponding probability of these events with the probabilities $p_i$ replaced by $\frac{1}{2}$ for all $1\leq i\leq n$?
Does one need some lower bound on the parameter $\eta_2$ so that the probabilities $q_i =1-p_i$ don't become arbitrarily small?
For the case where all the $p_i=p=\frac{1}{2} +\eta$, for all $1 \leq i\leq n$, with $\eta$ small enough, this is true with a basic calculation for the probability near the expectation value $np$, as $n\to \infty$.
Let $X_i, Y_i$ be independent Bernoulli random variables. Let $X_i\sim \text{Bernoulli}(p_i)$, $Y_i\sim \text{Bernoulli}(r_i)$. Let $\mu_n=\frac{1}{n}\sum_{i=1}^n X_i$ and $\nu_n=\frac{1}{n}\sum_{i=1}^n r_i$. Let $\overline{X}_n=\frac{1}{n}\sum_{i=1}^n X_i$ and $\overline{Y}_n=\frac{1}{n}\sum_{i=1}^n Y_i$. Suppose there exist numbers $0<a<b<1$ such that for all sufficiently large $n$, $\mu_n\geqslant b$ and $\nu_n\leqslant a$. Then for any $c\in (a,b)$, by the (independent but not identically distributed version of) the weak law of large numbers, $\mathbb{P}(\overline{X}_n\geqslant c)\to 1$ and $\mathbb{P}(\overline{Y}_n\geqslant c)\to 0$.
Your case satisfies this with $r_i=a=1/2$ and $b=1/2+\eta_1$. For a concrete choice of $c$, we have $$\mathbb{P}(\overline{X}_n\geqslant 1/2+\eta_1/2)\to 1,$$ $$\mathbb{P}(\overline{Y}_n\geqslant 1/2+\eta_1/2)\to 0.$$