Immeasurable sets are not something in the real world that you can throw a dart at.

I can rephrase your problem to be: "If I have an immeasurable set X in the unit interval, [0,1), and I generate a uniform random variable from that interval, what is the probability that that variable is in X?"

The problem is that a "uniform random variable" on a continuous interval is a more complicated concept than you think. Let me explain, by first giving an example where X is measurable, lets say X=[0,pi-3). We analyze random continuous variables by reducing to random discrete variables. We can think of a "uniform random variable" as a sequence of digits in a decimal expansion which are determined by rolling a 10 sided die. So for example, we can roll the die, and get 1,4,6,2,9,..., which would correspond to .14629..., which is not in the set X. Notice that while in principle we might have to roll the die arbitrarily many times, we actually only had to roll the die 3 times in this case, because once we got 1,4,6, we knew the number was too big to be in the set X. We can use this fact that we almost always have to roll the die only a finite number of times to get a definition of the "probability of being in X." In this case, we know that the probability is between .141 and .142, by considering 3 die rolls, and if we consider more die rolls, we get more accuracy that converges to a single number, pi-3.

Now, let's look at what goes wrong if X is not measurable. The problem here is that the set is so messy that even if we we know about the first finitely many digits of a random number, we wont be able to tell if the number is in X. This stops us from doing the procedure like above and defining what we mean.

Is this clear?