Let $Y_1,\ldots,Y_k$ be independent random non-negative integers and let $n>0$ be a positive integer. Does the following hold? $$ P(Y_1+\cdots +Y_k<n)\geq \prod_{i=1}^{k}\left(1-\frac{EY_i}{n}\right) $$
The case $k=1$ is Markov's inequality. I am actually mostly interested in the case when $k=n$ and the $Y_i$ are iid, although this generalization seemed to be true and might be easier to prove (for instance, by induction on $k$). Of course, it is possible that there is a simple counterexample...
Let $Y_1$ and $Y_2$ be iid. random variables such that both take the value $2$ with probability $p$ and the value $1$ with probability $1-p$. And let $n=4, \ k=2$.
Then
$$P(Y_1+Y_2<4)=(1-p)^2+2p(1-p)$$
and
$$E[Y_1]=E[Y_2]=2p+(1-p)=1+p.$$
The question is if
$$(1-p)^2+2p(1-p)\ge\left(1-\frac{1+p}4\right)^2$$ for all $p$.
But, no. For $p=\frac9{10}$ the left hand side is less than the right hand side. (The inequality turns upside down at $\sim 0.842$.)