RP

Remarks

Example: let $L$ be the set of triples $(A, B, C)$ of $n \times n$ matrices over $\F_2$ with $AB \ne C$. The machine picks a uniform $r \in \F_2^n$ and accepts if $A(Br) \ne Cr$. If $AB = C$, it never accepts; if $AB \ne C$, the matrix $D = AB - C$ has a nonzero row, so $Dr \ne 0$ for at least half of the vectors $r$. So $L$ is in RP.

Non-example: a machine that accepts some $x \notin L$ with probability $1/100$ fails clause 3, even if its error is small. Such two-sided error is allowed in BPP but not in RP.

The constant $1/2$ in clause 2 can be replaced by any constant in $(0, 1)$, or by $1 - 2^{-q(n)}$ for a polynomial $q$, by independent repetition. Clauses 2 and 3 show that an accepting run exists exactly for $x \in L$, so an accepting sequence of coin flips is a certificate of membership and $\mathrm{RP} \subseteq \mathrm{NP}$.