# Probability of picking a real number randomly

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?

• nice question. A bit of limit based approach will help. (something like the probability tends to 0) Jan 9, 2021 at 10:16
• Also, notice that when numbers get infinite, we move on to an higher dimension. here from 1d to 2d. rather than probability of a single point on a number line, we now talk about a finite interval (line segment), whose probability can be calculated by measuring the 'length' Jan 9, 2021 at 10:18

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.

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 $$0$$.)

• Well, yes and no...in real life, the coin flip thing will never happen. But in the OP's setup, we know that something with zero probability is bound to happen. I would say the resolution is that in real life, we can never pick a real number randomly. Nov 23, 2020 at 19:53
• @TonyK It will almost surely never happen. That's what probability 0 implies. I see no mathematical reason why every single coin flip ever done from this moment forth can't be a tails. It is just as likely as any other particular sequence of throws. Nov 23, 2020 at 19:55
• @TonyK In addition: If we keep throwing coins indefinitely regardless of the results along the way, then the sequence we work our way towards will have a prior probability 0 of occurring. In my setup I just choose to stop when it is clear that the sequence we get isn't the sequence I guessed ahead of time, which is "just tails, from now and forever". Nov 23, 2020 at 19:58
• But Arthur, life is too short to toss an infinite number of coins. That is why I said in real life we can't pick a real number randomly. Nov 23, 2020 at 20:03
• @TonyK When has math problems ever cared about such minute details as "real life"? Nov 23, 2020 at 20:03