In a continuous distribution the probability of any number on that distribution being generated is effectively zero. If R was generating the true probabilities it should give you zero for every single number.

Think about it. That distribution is continuous over an infinite amount of numbers. If you select any number the chances of that number being generated will be essentially zero. According to the theory there is no possibility for any number on the distribution to be generated. This is correct.

Yet when you use the random number generator you get a number even though that number technically is impossible to exist due to zero probability. Does this mean there is a flaw in the theory when applied to the number generated?

Yes it does. The theory is an approximation of what's going on itself. No random number generator in a computer is selecting a number from a truly continuous set of numbers. It is selecting it from a finite set of numbers from all available numbers in a floating point specification.

Even if it's not a computer when you select a random number by intuition from a continuous distribution you are not doing it randomly.

Think about it. Pick a random number between 0 and 1. I pick 0.343445434. This selection is far from random. It is biased because there is an infinite amount of significant figures yet I arbitrarily don't go past a certain amount. I cut off at 9 sigfigs and bias towards a cutoff like that because picking a random number with say 6000 sigfigs is just too inconvenient. You really need to account for infinite sigfigs for the number to be truly random which is impossible.

So even when you pick numbers randomly you are actually picking from a finite set.

In fact I can't think of anything in reality that can truly be accurately described with a continuous distribution. Nothing is in the end truly continuous. Or maybe it does exist, but if it does exist how can we even confirm it? We can't verify anything in reality to a level of infinite sig figs.

If R was accurately calculating likelihood it should give you zero for each number. And the random number generator should not even be able to exist as how do even create a pool of infinite possibilities to select from? Likely R is giving some probability over a small interval of numbers.

That's where the practicality of the continuous distribution makes sense when you measure the probability of a range of values. You get a solid number in this case.

Anyway the above explanation is probably too deep. A more practical way of thinking about this is like this:

It is unlikely for any one person to win the lottery. Yet someone always wins. The probability of someone winning is 100 percent. The probability of a specific someone winning is 1 over the total number of people playing.

Improbable events in the universe happen all the time because that all that's available. It's highly improbable for any one person to win the lottery but if someone has to win, then there is a 100 percent chance that an arbitrary improbable event will occur.

This is more easily seen in a uniform discrete distribution rather then the normal continuous distribution.

In the case of the normal distribution it is confusing. In a normal distribution It is far more likely for an improbable event to occur then it is for the single most probable event to occur.

Think of it like this. I have a raffle. There are 2 billion participants. Each person has one ticket in the bag, except me. I have 100,000 tickets in the bag I am the most likely person to win.

But it is still far more likely for anyone else but me to win even when I am the most likely person to win. An arbitrary improbable event is more likely to occur then the single most probable event.