Technically speaking, no. Real analysis is pretty self-contained. You basically start out by constructing the reals from scratch and deducing continuity as a consequence of the completeness of the real field. You use a tiny bit of set theory to establish notation and define bounds, and then from there you go into limits, derivatives, integrals and maybe the Lebesgue measure. I wouldn't expect a real analysis course targeted at applied math majors to do much other than that.

The reason real analysis is useful is because it's (loosely) a deeper calculus course with proofs. Since probability theory becomes more proof-based (and ventures into measures), real analysis is good preparation for it.