Do RP algorithms exist?
The algorithms you are looking for are also called one-sided-error Monte Carlo algorithms.

The idea is to randomly guess a witness for the input being a YES-instance. If you find one, your answer "YES" is always correct; if you don't find one, you say "NO" even though the correct answer may be "YES" (we want to restrict the likelihood of this event). If the input is a NO-instance we can never find such a witness, so we always output NO correctly.

One well-known example is the Solovay–Strassen primality test.