I think it's true because you can reduce B to A and then run the probabilistic algorithm of A and if we got a reject then it's also a reject for B and at least half of the time if the input is in A we would get an accept which means that at least half of the time if the input is in B we would get an accept. So B is in RP, but I'm not sure.
What happens if the reduction is the other way around? If A is in RP and there is a polynomial time reduction from A to B, then is B in RP?