Comparing a Poisson binomial distribution to a classical binomial distribution

Question

Let $X$ be a random variable that follows a Poisson binomial distribution of parameters $p_1,...,p_n$, and let $p$ be such that $p_i<p$ for all $i$, and let $Y$ be a random variable that follows the classical binomial distribution $B(n,p)$. What's the easiest way to prove the very intuitive result that for all $k>pn$, we have $P[X=k] < P[Y=k]?$

Answer 1

Thanks to @Michael Lugo for the comment — the concept I was looking for was indeed coupling arguments: the "Biased coin" example on that Wikipedia page is exactly what I needed here.