You are intermixing two things:
- something is random or not
- the distribution of something, which might be equal or not¹
¹) very often, it isn't an equal distribution, but a normal distribution, when most cases are grouped around an average. Think: body size of persons of same age and culture: Most are around 1.8m, fewer 1.7m or 1.9m, far fewer 1.6 or 2.m. But rarely two childs from the same parents are of exactly the same length.
If you enter a casino, you will hardly find anybody who can predict the next card from the deck, or the next roulette number. Some might believe they know a way to predict it, but they can't.
Some people here might think, that if they knew enough parameters, they could predict the next roulette number, but think of it! The microscopic shape of the billiard ball will change with the temperature, and you can't shield a Casino from the outside world, to prevent any influence. The air is moved by the breathing of the visitors, and you can't calculate when whom of them will breath - it's not a problem which will be solved in 100 years with more computer power. It is systematic: The finer conditions of roulette are uncontrolled.
So the outcome is random.
My first impulse was, to write "true random", but as serious adults, we always tell the truth, of course, so if we say random, we mean random.
You might construct a roulette canon in a laboratory, which generates predictable results, but you will not get a license for that. That much is predictable too. :)
Now to the second, related questions. Some people believe that only equal distributions, like from throwing a dice, are random. But that's not the case.
Consider a bowl with two black and one white ball, and you pull blindly from the bowl and put it back each time. You will pull a black ball with 2/3 chance and a white one with 1/3 chance. It is not predictable which color you will get, but the probability is not equal - it is 2:1.
But knowing that, you could transform it into an equal distribution: If you pull a white, it is white, but if you pull a black, you have to pull a second black one. Generated by a PRNG:
w, b, b, w, b, w, w, b, b, b, b, b, b, b, b, b, w, b, b, b, w...
W B W W W B B B B W B
Here we see a 5:6 relation, which is of course no proof, but the proof is more a case for Mathematics.SE or Statistics.SE. It's only to show how it works. Now: If you know the bias of any random distribution, you can transform it into an equal distribution, which means, that an equal distribution isn't such a magical thing. You could for example measure red and non-red cars which appear on the other side. If you counted a lot of them, and know the distribution, you can correct the unequalness by calculation.
People with a deterministic viewpoint might argue, that, given enough data and big machines, and knowing all natural laws in greatest detail would make it possible, to calculate every minor event in the far future, but we know that there are far too many atoms in the universe, where we don't know where they are, and what their movement is. By all practible means it is impossible to predict the future, but that's what random is about: Can we predict it or not. It's not about "Could it be predicted, if ...".
So the statement, that our future isn't predictable doesn't need a proof - it is a fact. A deterministic viewpoint is, what would have needed a proof, but from quantum mechanics we know, that there is uncertainty on the subatomic level (Heisenberg).