But at some point in understanding probability, as it is actually used in the real world, you need calculus.At SOME point, true. But that point is very, very far off, and very few people ever get there.
It is possible to understand what the normal distribution is, and how to lookup significance values, without worrying about how to calculate it. This is how virtually every non-statistician does it, and is what statisticians themselves do more of the time.
The point where you have to calculate it and prove properties about it does require Calculus.
Unfortunately, calc generally requires trig. It should be taught while teaching calculus.
I'm sorry, but this is not a good idea. What Calculus requires from trig is a solid understanding of what sin, cos, and tan are (else the derivatives of the same won't make sense), and a solid understanding of trig identities for use in integration. Both uses require several layers of abstraction on top of the idea of trig. My experience is that it is a bad idea to layer abstractions on top of material before making sure that that material is solid.
I therefore want people to have a solid understanding of trig before they arrive in a Calculus course.
That said, I think that a lot of the use of trig in integration is a now useless skill, given the widespread availability of programs like Mathematica that can solve all of those problems very quickly and much more accurately. It was once important for people to learn those skills, but now it doesn't seem that useful to me.
You should be able to verify that the integral differentiates correctly. But integrating complicated expressions isn't in my view that critical of a skill.