Probability of at least two of n independent events occurring subject to some conditions

Question

Given a set of independent Bernoulli random variables $\{x_1, \dots x_n\}$, let $p = \sum_{0<i\leq n}\Pr[x_i = 1]$ and $X=\sum_{0<i\leq n} x_i$. We know that for any $i$, we have $\Pr[x_i = 1]\leq \frac{p}{2}$ and need to find a lower-bound for $\Pr[X>1]$ for any $p \in [0, 2]$. (The lower-bound should be a function of $p$.)