But what about the process that produces those distributions? Is that process called random?
If I throw a loaded die that tends to produce 4s and 6s and rarely produces 1,2,3,5 is that considered 'random' since the it produces a distribution? I understand that you speak of the distribution as random, but what of the process that produces that distribution?
Here's where we get into confusion of the technical definition vs. the layman's definition. I'll answer for the technical definition.
Yes, the process is random. Furthermore, I want to be quite emphatic about that. This isn't my opinion. It's the opinion of every probability textbook ever written.
The definition of a random process is one which is described with random variables.
(Alternate definition which is even more accurate: An ordered sequence of random variables. That definition is really weird though, because despite its accuracy, it defines a "random process" without referencing anything about a "process". There's good reason for doing it that way, but it confuses a lot of people, and I won't discuss it any further unless someone asks.)
Now for the confusing part.
A lot of the time, people use the word "randomly" to refer to "without bias". If I pick a card "at random", every card is as likely to come up as any other.
Sometimes, this is confused with "all probabilities being equal", which isn't correct, either. We think of it that way because we have an image of a die as having six choices, and we expect each of those six sides to come up as often as any other. So, an unbiased, i.e. random, selection will also have a uniform distribution.
However, some games call for something called a "average die". It has six sides, and they are labelled 2,3,3 ,4,4,5. The value rolled "randomly", i.e. without bias, will not have a uniform distribution on the numbers 1-6. It is still a random number, though.
When we think about loaded dice, why do we say that they are not random? From a technical definition, they are random, but with a nonuniform distribution. However, when we think about dice, we expect them to produce uniform distributions, and we create games of chance based on the fact that they will do so. When someone knowingly uses loaded dice, they are probably cheating. In other words, they are violating the "unbiased" sense of the word "random". They are deliberately fixing the game so that the numbers are selected using a distribution favorable to them, instead of the expected distribution. We often say that the numbers produced were not random, because they were not the random numbers expected if an unbiased process were used.
Even in probability textbooks, you'll sometimes see that definition used in the examples. If someone says that a random number is generated rolling a fair die, we assume that what they mean is a random number whose values are integers uniformly distributed on 1-6. That isn't the definition of "random", though, that's the definition of a "fair die".
