Choosing something of $0$ probability

Question

First of all, i am only a newbie and i am pretty sure that my thinking is faulty somewhere, so along the way as i explain things i will probably say something with an error and i hope you will help me find this error. This idea has been bogging me for some time already.

Imagine picking a random real number. Without the loss of generality, let the picking interval be $(0,1)$, or $<0,1>$.

Now, since there is an infinite amount of real numbers, the probability of choosing one specific number is $0$.

Yet, we have to pick one number when choosing randomly, let's say it's $0.5$. We have now picked a number that had $0$ chance to be picked.

How is this possible?

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Thank you for all the answers, i have had hard time deciding what to pick as "correct" answer as i came seeking answers.

EDIT: Recently, i saw these Binomial Distributions video series and this video explains it pretty well

Answer 1

What you're looking at here is the difference between impossible and probability zero. An impossible event is one which literally cannot happen; for instance, choosing a random number between $0$ and $1$ and getting a result of $5$. These events aren't even conceivable within the universe of discussion.

A probability $0$ event is one which is conceivable within the universe we're discussing, but which has no positive likelihood of occuring. Your example -- choosing a random number in $[0,1]$ and getting $0.5$ -- is a great one. Another good example is the event that a coin is flipped, then flipped again, then again... and so on forever, and always ends up heads. It could happen... but in (literally) all probability, it won't.

Answer 2

Remember that mathematics deals with an idealized model of the world. In reality you cannot observe an event that has an infinite (let alone uncountable) number of possible outcomes; even if the underlying physics of your random process is truly continuous, you cannot measure the $0.5$ outcome exactly. You might only be able to say for sure that the outcome was between, say, $0.499999$ and $0.500001$. And the event that $0.499999 < X < 0.500001$ where $X$ is uniformly distributed over $(0,1)$ has a non-zero probability.

In the idealized mathematical model, on the other hand, probability measure of a continuous random variable is a matter of integral calculus. The probability of any one exact observation is zero (the result of multiplying a zero-width interval by the pdf at that point), but the total probability over all the observations in a range is the integral of the pdf over that range, which is non-zero.

Answer 3

The probability is infinitesimal, there's a wikipedia page about the the opposite case with probability 1 here.

Answer 4

In the real world, once you have picked a number, the probability that you have picked it is $1$, never anything else. Secondly, it is impossible to have a process to pick a number from $[0,1]$ uniformly at random. For one thing, you can only write down a finite number of digits, so any process that terminates in finite time can only produce finitely many possible numbers, in which case any number that has zero probability of being produced will indeed never be produced. Also, if you say that the random number is simply the position of some particle, that is totally invalid because no particle is at only one point. (In fact every particle has a wave-function that extends throughout the universe.) Thirdly, it is not known or provable whether there is any true source of randomness in the universe!

Answer 5

I too am a newbie but I think the distinction that you are after is the difference between probability 0 and probability approaching 0.

Like the probability of a coin heads or tails and the result is it lands on the edge or you lose the coin. Those aren't taken into account in the original equation but are slight possibilities.