How does one change the probability bounds in probabilistic complexity classes without changing the class?
Suppose that $\Pr[M(x) = L(x)] \geq 1/2 + \epsilon$. Run $M$ repeatedly $m$ times (for $m$ odd), and take the majority vote. Let $x_i$ be the indicator for the event that the $i$th run of $M$ gives the correct solution. The success probability of the majority vote algorithm is
$$
\Pr[x_1+\cdots+x_m &gt; m/2] \geq \Pr[\operatorname{Bin}(m,1/2+\epsilon)&gt;m/2].
$$
At this point you apply the Chernoff&ndash;Hoeffding bound. You take it from here.