> So in a sense, the huge body of mathematics underlying probability has already failed us at this basic juncture because we cannot speak of how random one particular outcome of an experiment is.<p>This reminds me of the 'Is 14 a random number?' debate.<p>Part of the problem, though, is that we have terminology that is horribly misleading.<p>First, a random variable is neither random nor a variable. They're not variables, because they're functions that yield non-deterministic values - a <i>huge</i> distinction. You can't have a random number - the idea itself doesn't make any sense.<p>Second, the definition of 'random' is itself problematic - or at least contested. Randomness implies probability, but probability can be defined in two different (and incompatible) ways, one of which is essentially the inverse of the other. Ironically, the one that Keynes proved back in 1921 to be logically inconsistent is the one that is more commonly used today (probably because it is more intuitive and mathematically convenient... even if it is almost always incorrect!).<p>The question 'Is 14 a random number' doesn't really make any sense. 14 cannot be a random number; it can only be a number drawn from a random distribution. That may seem like it simply begs the question, but in fact, this subtle shift is incredibly important - determining whether a number was drawn from a random distribution is much easier to frame in terms of probability, and probability, not randomness, is the language of statistics.<p>Unfortunately, this is one of those cases where the two definitions of probability yield widely different answers. You could tell me that the answer is undefined, in almost exactly the same way that division-by-zero is undefined in mathematics. Or you could construct a model over all possible distributions of numbers, the probabilities associated with each of those distributions, and integrate accordingly to yield some (probably computationally unfeasible) functional answer.<p>In this school of thought, we <i>can</i> speak of how 'random' a particular outcome is - we're essentially partitioning the (potentially infinite) universe of functions <i>f</i> such that our value <i>v</i> is in the range of <i>f</i> into two categories: one designated as 'random' and the other designated as 'not random'. Then, we are determining the probability that our value was generated from one of the former, as opposed to the latter.<p>Either one would be correct ways of answering this second question, but neither one addresses the first question, which is essentially nonsensical. (Well, I guess the answer is 'no, 14 is not a random number, because a number cannot be random', but that's a bit of a cop-out!).