On the Probability a Weighted Bernoulli Sum Exceeds Its Mean

Let $w_1,\dots ,w_m$ be positive real weights whose sum is $1$, and let $v_1,\dots ,v_m$ be i.i.d. Bernoulli$(p)$ random variables. If we let $X={\sum }_{i=1}^m{w}_i{v}_i$, then we conjecture that for all $0\le p\le 1\mathrm{/}3$ we have $$\mathbb{P}[X\ge \mathbb{E}[X]]\ge p\mathrm{.}$$ In this short note, we observe a connection of this conjecture with a version of the Manickam-Miklós-Singhi conjecture, which allows one to prove it for sufficiently small values of $p$.