Probability in Measuring Noncommuting Observables 
If I have a particle in a state $\Psi(x) = e^{-x^2}$ could I calculate probability of simultaneously measuring, say, $x > 0, p_x < 0$? 

I understand that $p_x$ and $x$ don't commute and cannot be measured simultaneously with arbitrary accuracy, but does that also mean I can't determine the probability of measuring them in these ranges? Thanks for any help.
 A: You could use a Wigner distribution which does generate positive numbers for regions of phase space that are large enough.
But when operators don't commute it also means there is no experimental setup that can evolve them into a state that would give the same result repeatedly if performed again and again.
So you've jumped over the whole fact that a so called measurement gives a result (with a particular frequency for an ensemble and with repeatedly getting the same result when repeated on the same object). There is no result in your setup, just a frequency (and only then if you use the Wigner "distribution," which isn't a distribution since it goes negative) and then there is no clear connection to any experiment.

I understand that $p_x$ and $x$ don't commute and cannot be measured simultaneously with arbitrary accuracy

They can't be "measured" simultaneously at all. And if you switch back and forth, measuring one then the other, then the so called measurements will be changing the state each and every time.

but does that also mean I can't determine the probability of measuring them in these ranges? Thanks for any help.

You could measure one, then measure the other but then if you go back and measure the first it is possible to get a different result. If you had measured the position twice then the momentum the two position results would agree with each other. If you put the momentum in between then the two position measurements can give different results. The momentum measurement has clearly changed the state (it used to be a state that gave particular position results, now it isn't).
So even if you tried to make a third kind of measurement one that puts it into a new state one that one gives positions in one region of position space and only gives momentums in another region of momentum space then you have to specify all the other possibilities before you've made something that can be measured.
And for some potentials it is possible. For instance if you have a potential that is infinite when $x>0$ then a momentum eigenstate with momentum of the correct sign could be what you want and gives 100% as the probability. But momentum eigenstates can't be made in the lab and neither can that potential.

This makes the issue really clear, but I was wondering, I can easily look at the wave function in position and momentum space separately and easily calculate the probabilities which disregards the simultaneity of the measurement, so would the error become apparent if I were to calculate the simultaneous probabilities using a wave function in phase space?

There isn't a position and momentum just sitting there with a probability of having a range of values. After you decide on a measurement to do, you can compute the probability of creating certain results. But when you compute the probability of creating a certain momentum result and compute the probability of creating a certain position result you are computing probabilities of different universes, one were you create momentum results and one where you create position results. You most certainly are not revealing preexisting facts since first creating a position result, then creating a position result, and then finally creating a momentum result gives different correlations between the three results than first creating a position result, then creating a momentum result, and then finally then creating a position result.
Consider a situation where you compute the probability of carving a tree into a coin, burning the rest and then flipping it, it would be 50-50. Or you could compute the probability of carving a tree into a single die and then burning the rest and then rolling it, the probability would be 1/6 for each outcome. You can compute or do either. But the idea of simultaneously doing both just isn't a thing. And computing the probability of each means nothing about doing both since you can't do both.
When you create position results you interact with one kind of device. When you create momentum results you interact with a different device.
