Definition. Amplification [amplification]

Modifying the definition of $\text{RP}$ a little bit: $L \in \text{RP}(p)$ if there is a PTM that, on input $x$ , runs in time $\text{poly}\left( |x| \right)$ and has behavior:

$\text{Pr}\left( \text{reaching accept} \right) \geq p$
$\text{Pr}\left( \text{reaching reject} \right) = 1$

Example. Claim: $\text{RP}\left( \frac{1}{2} \right) = \text{RP}\left( \frac{3}{4} \right)$

Proof.

Let $L \in \text{RP}\left( \frac{1}{2} \right)$ . There exists PTM $M$ such that $x \in L:P\left( {M\ \text{accept}\ x} \right) \geq \frac{1}{2}$ , $x \notin L:\text{Pr}\left( {M\ \text{rejects}\ x} \right) = 1$ .

Let $N$ be the Turing machine on input $x$

Run $M$ on input $x$
Run $M$ on input $x$
if either accepted, then accept
else reject