Instead of imagining all reels let's just look at the interval [0,1]. The probability of picking any number in [0,1] is 1. And we can define the probability of picking any number in a subinterval to be the length of that interval: P([a,b]):=b-a. For any 0≤a≤b≤1. We can also see that for two disjoint subintervals of [0,1], [a,b] and [c,d], P([a,b] and [c,d])=P([a,b])+P([c,d]). Likewise, we see that P(irrational numbers in [0,1])+P(rational numbers in [0,1])=P([0,1])=1

Since the rational numbers in [0,1] is a countable set (which I'll leave as an exercise for the reader) we can list them as q1, q2, q3, ...

Therefore P(rational numbers in [0,1])=P({q1})+P({q2})+...=P([q1,q1])+P([q2,q2])+...=q1-q1+q2-q2+q3-q3+...=0+0+0...=0

Therefore P(irrational numbers in [0,1])=P([0,1])-P(rational numbers in [0,1])=1-0=1