= Solution
Write $N_m=|B(m)|$ and $\theta=\theta(p)>0$. By translation invariance and <linearity of expectation>,
$$
\mathbb E_pR(m)=\sum_{x\in B(m)}\mathbb P_p(x\text{ belongs to the infinite cluster})=\theta N_m.
$$
Let $r=\mathbb P_p(R(m)\geq\theta N_m/2)$. Since $0\leq R(m)\leq N_m$, splitting the <expectation> at the threshold gives
$$
\theta N_m\leq\frac{\theta N_m}{2}(1-r)+N_mr.
$$
Rearranging produces the slightly stronger estimate
$$
\boxed{\mathbb P_p\left(R(m)\geq\frac{\theta(p)|B(m)|}{2}\right)\geq\frac{\theta(p)}{2-\theta(p)}\geq\frac{\theta(p)}2.}
$$
Only the mean and boundedness of the <random variable> are used; independence of the vertex-membership events is unnecessary.
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