I'm definitely not at all qualified to talk about this, but... aren't "uniform" and "random" antonyms?
https://numpy.org/doc/stable/reference/random/generated/nump...
Or in god's own words (TAOCP section 3.4):
Applications of random numbers often call for other kinds of distributions, however; for example, if we want to make a random choice from among k alternatives, we want a random integer between 1 and k. If some simulation process calls for a random waiting time between occurrences of independent events, a random number with the exponential distribution is desired. Sometimes we don't even want random numbers — we want a random permutation (a random arrangement of n objects) or a random combination (a random choice of k objects from a collection of n).
In principle, any of these other random quantities can be obtained from the uniform deviates U0, U1, U2, ...; people have devised a number of important "random tricks" for the efficient transformation of uniform deviates. A study of these techniques also gives us insight into the proper use of random numbers in any Monte Carlo application.
The easiest example most people are aware of in practice is throwing two 6 sided dice vs one 12 sided dice. The outcome of both is random, but people seem to know fairly intuitively that the two dice case produces is more likely to produce middle values than extreme ones, while the single 12 sided dice doesn't have that behavior.
My reasoning is, if you are counting cards in a game, you are giving yourself an advantage. But what really happens is that from your point of view cards will be drawn less and less uniformly randomly. Or put in another way, if you know the distribution is normal, you can bet on the the result being near the center and come out on top, but if it’s uniformly random distribution all hopes are out.
The problem is if you are doing something that requires uniform random values, and your random source is not uniform, then things will go wrong.
Finite support somehow seems less random because you can still say something interesting about its mean and bounds in the same way you can say something about the mean and variance of a normal distribution.
Even more so if you’ve gone 50 picks like that, and are now down to 49 red balls and one blue ball.
That is what card counting helps you with - knowing the odds based on your current state, as compared to the initial state.
And the last card was still random, btw. But can be guessed now with perfect precision due to the process of elimination.
Random at shuffle doesn’t mean unguessable or unpredictable as the game goes on.
I disagree the last card is "still" random, at least in the sense of statistics. It "was" random up until the measure of uncertainty of the next event reaches 0 (i.e. entropy reaches 0). At that point it's no longer a "guess", there is no uncertainty and the remaining pattern is always 100% predictable in that regardless which proceeding events occured to get there it can always be known what the next value is without uncertainty. Since there ceases to be any uncertainty in what the remaining pattern will be there ceases to be randomness in the next value generated. That the card's value was not known at n=0 does not affect whether the n+1 card still is/isn't random when n=51. In another form, that you didn't previously know the value of card n=52 with past information holds no influence whether the value is random or not with new information. Statistical randomness is all about what you know of the future predictability, not about how something came about.
This is also true of events which fall into predictable patterns at any point along the path. E.g. if I had a (relatively useless) hardware random number generator that generated random numbers 0-127 once per second until it generated a 0 at n=17, at which point it ceased being able to pull randomness from it's dead circuits and always produced 0 afterward, the first 17 values were all statistically random at the time of their draw but n=[18,inf) are all now predictable and no longer random from that point on.
Knowing the outcome of a die roll after the roll doesn’t make the roll of the die non-random.
Any more than the die coming up 6 a bunch of times in a row does. Though maybe it would be a good idea to check the weight/balance of it, hah.
Knowing what the last card has to be after all the other cards have been played doesn’t make the process which made that card the last card any less random either.
Picking cards off a deck isn’t a randomization/non-randomization event, shuffling the deck is what does that.
Probability however, which directly impacts gameplay in some kinds of games, is directly impacted by knowledge exposed during gameplay.
So for instance, the second to last card (and also the last card) can be known with about 50/50 odds (probability wise) if someone is good at counting, which is damn good! Way better than 1/2704 odds of guessing a sequence of two cards at the start of the deck.
In the example of the broken hardware random number generator, knowledge of the defect can be used to attack a system if it assumes a continuous probability distribution out it’s outputs, if the attacker knows this.
The same as an attacker at a casino could probably manipulate a game to make money if they knew something was broken in the deck shuffling and the same two cards were always at some position in a ‘new’ deck.
Exactly, same page then. The card not being random at the end of game frame of reference places no limitation that it was random from the beginning of the game frame of reference. In the end of game reference the last card is never random, it's only ever the remaining card. In the beginning game reference which card will end up being the last card is still random at that point despite it ceasing to be from the later frame.
Rolling two dice gives you a number which is random in [2, 12], but not uniform -- 7 is far more likely than 2 or 12.
But what if the domain isn't finite? e.g. the entire real line? Then we can't define a uniform distribution, because the total probability can never sum to one. We can still define a maximum entropy distribution, but only by making some more assumptions. The normal distribution is the maximum entropy distribution on the real line, given a particular mean and variance.
Other examples: on the positive real line, the exponential distribution is maximum-entropy for a given mean. Poisson distribution is the equivalent on the non-negative integers.
That’s just because the payout distribution and real distribution are different, and there’s a place the payout is at a loss.
If you bet on a uniform distribution of 2 through 12 that payed out according to a normal distribution, you’d make a killing betting on 2 or 12 (the “outliers”). If you roll 2d6 (normalish) and get paid according to a uniform distribution, you’ll make a killing betting on 7.