At some point (depending on your textbook or prof) in applied maths, you can't do prob/stats without calculus. As to the fundamental theorem in calculus, remember that it took Reimann and analysis to finally prove it.
In the real world (which I have little exposure to), stats are probably much more important. But in my little version of reality, Runge Kutta solvers and ODEs show up as often as the central limit theorem. My focus is on applied maths, however, so I see a wide variety of methods from the world of probability and statistics, as well as graph theory, plus Fourier analysis (calc required, of course), finite element analysis, PSO, etc.
I'm still waiting for neurosciecne to pin down how the human brain performs the ODEs involved in throwing or catching a ball. I doubt there's any ODEs involved; instead, the brain just uses some heuristic that evolved to be good enough in the real world.