I don't know if there are 'natural' events with irrational probabilities, but I think we can construct an artificial one? Roughly, let X be the event that a uniformly-random number in [0,10) is < pi; then P(X)=pi/10.* So if you were forced to give X a rational probability, what probability would you choose?

*One can't actually produce a uniformly-random real number, but here's a real-world procedure that should be equivalent to X: start generating an infinite string of digits v_i one at a time using a d10. For each digit, if it is less than the i'th digit of pi return YES, if it is greater return NO, if it is equal continue to rolling the next digit. So e.g. a roll of 2 is an immediate YES; a roll of 3 -> 1 -> 7 is a NO. This procedure takes unbounded time, but with probability 1 it terminates eventually.

Cool. So in principle we could just as well use the rationals from the standpoint of scientific inference. But we use the reals because it makes the math easier. Thank you.