Not much. The kind of probability relevant to MIRI's interests is not the kind of probability you need calculus to understand (the random variables are usually discrete, etc.). The closest thing to needing a calculus background is maybe numerical analysis (I suspect it would be helpful to at least have the intuition that derivatives measure the sensitivity of a function to changes in its input), but even then I think that's more algorithms. Not an expert on numerical analysis by any means, though.

If you have a general interest in mathematics, I still recommend that you learn some calculus because it's an important foundation for other parts of mathematics and because people, when explaining things to you, will often assume that you know calculus after a certain point and use that as a jumping-off point.