Probability of picking a real number randomly

Question

If we randomly pick a real number from the number line, the probability of picking a number (say x) is 0. This is true for all real numbers x and it makes sense to me why this must be true. But seemingly there is a paradox lurking here. Suppose we pick the number y. This number also had probability 0 but was still chosen in the random process. If it was chosen, it means the probability of it being chosen was finite. Isn't this a paradox? What am I missing here?

Answer 1

Infinity doesn't work like you think it does. There are several ways to interpret probability. One of those ways is the "frequentist" interpretation - you divide the number of outcomes you're looking for by the number of outcomes possible, and that's the probability. The issue is that, although that's fine for finite sets of outcomes, it doesn't work when there are infinitely many outcomes (or, more accurately, it always produces zero).

In your interpretation, you are really asking "what is the probability that the number is $y$ now that I've already selected it, and the answer is 100%. That isn't surprising - there is one outcome and you've chosen it.

However, from the frequentist perspective, what was the probability that you would select $y$ and not, say, $y + \delta$, for any real $\delta$? Well, there are two issues:

1. There are infinitely many reals on any interval, so technically this is $\frac{y}{\infty}$, which is always zero

2. Worse, the size of the interval doesn't seem to matter. Whether we choose some extremely small interval or an extremely large one, the size is still infinite.

It is #2 above that breaks the frequentist interpretation entirely unless the probability of a particular outcome is always zero. Basically, the probability of a particular outcome in the frequentist interpretation must change (or stay at zero and only zero) if the set of possible outcomes expands. Specifically it must reduce, presuming that the outcomes are independent. You cannot have a probability less than 0, so it works out.

So instead we don't talk about probability of a given number. We talk about the probability that the outcome will be within some interval, which does have a very well defined interpretation from the perspective of the size of the various sets.

Answer 2

Probability zero does not mean impossible. Just as probability 1 doesn't mean guaranteed to happen.

A different example of the same phenomenon is this: Flip a coin until you get a heads. The probability that you get a heads at some point and therefore stop is 1. But it clearly isn't completely, absolutely guaranteed to happen. The probability that you keep getting tails indefinitely and never stop is 0. But it clearly isn't impossible. (This is actually almost equivalent to picking a real number uniformly at random from the interval $[0,1)$ and asking whether the chosen number was $0$.)