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.
> You basically start out by constructing the reals from scratch
Not necessarily -- a lot of books just take the existence of a unique set with certain properties as an axiom and call it R.
The main topics of basic real analysis IMO are differentiation, integration, (uniform) continuity, compactness, convergence, etc.; how to construct the reals from the rationals is a side point at most.
This could be personal bias as I just don't personally think that the exercise of constructing the real numbers is very interesting.
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.