## Nothing is impossible (?)

Something’s been bothering me for a while now. While my statistical intuition is usually quite sharp, in this case I’m wrong, although I never really heard a convincing argument why exactly. So I’ll try to explain my problem, and hopefully someone with a sound argument could reply. Basically, my confusion starts with the sentence ‘Some event that has a chance of $0$ of happening, can still happen.’ Let $P(x)$ be the chance that $x$ happens. To me `$P(x) = 0$‘ is equivalent to the statement: ‘We are certain that $x$ will not be the case’. So saying that $P(x) = 0$ doesn’t exclude $x$ from happening feels just like a contradictio in terminis, to me. For example, the chance that you throw a (normal) die and a $7$ turns up*. But this is a silly example, because everybody agrees that the chance of this happening is $0$ and also that it is impossible. But now, consider the following: If we pick a random number, the chance we pick a certain number $N$ is clearly $0$. Since this holds for every $N$, you might argue that this implies I’m wrong, because apparently something happened which had a chance of $0$. To me this just implies that it’s not possible to choose a number randomly. A different example: we flip a coin until heads turns up. Is it possible that this game never ends? To me, it’s not. Of course, it could take arbitrarily long, but it can’t take an infinite number of flips. Can it?

While trying to emphatize with the people who believe that ‘impossible events are possible’, I came up with the following: let’s say we pick a random real number from the unit interval. Since there are uncountably many reals and only a countable number of rationals, the chance to pick a fraction is $0$. But you could still say: ‘well, it’s only random. And fractions actually do exist in abundance. So there is no obvious way that it shouldn’t be possible to choose a fraction by accident’. While this idea makes sense to some extent, I believe it’s wrongheaded. Mostly because, like I said above, this just seems to be based on the belief that it is actually possible to pick a number in some truly random way.

In any case, this is not meant as a convincing argument against the idea of the impossible becoming possible. I’m just trying to show where I get confused, and I am in need of some explanation. And after all, I’ve never seen anything in my life happen that had a zero chance of happening. So the burden of proof lies with the people who claim that $2$ is a random integer ;).

*maybe it actually could happen by some weird quantum effect, but mathematically speaking, it’s not an issue

PS. Terry Tao is going to set up another mini-Polymath Project next month. Check it out.

### 4 Responses to “Nothing is impossible (?)”

1. nhung Says:

I like this blog very much^_^.

2. My idea Says:

😉

3. My idea Says:

;;) :)) :((

4. My idea Says:

:-b :-d

While I clearly appreciate your smilies, I have to ask you to make your comments slightly more senseful in the future, b:-d). W