You are given a machine made by scientists that they say is perfectly random. All the machine does is display ten numbers every year on the same day. But the machine has just been made. So on its very first display of numbers, it lists out ten 5's. Do we assume it's random or assume the scientists made a mistake, in which case the machine is deterministic? — Gregory

This is too easy. A "machine made by scientists" is a physical device. It therefore has unavoidable imprecisions in its manufacture; and therefore it's not perfectly random, refuting your assumption hence your argument.

The only way to salvage this is to say we have a "machine made by God." Which, by way of example, might as well be taken as the universe. Then I flip a fair coin and it comes up heads a billion times in a row. Is that evidence of the failure of randomness?

Suppose the coins come up in any pattern whatsoever. Isn't the probability that they came up in that exact pattern also $\frac{1}{2^{10^9}}$? Wouldn't you take THAT as a failure of randomness?

We are forced to conclude that anything that happens at all is so unlikely that it counts as proof that there is no randomness. The big bang happens. Stars form. Our solar system forms. Eons pass. Life appears on earth. Millions more years pass. Humans crawl out of caves and build civilization. Your parents meet and produce you; and here you sit at your computer or phone and read my words. What is the prior probability of THAT?

Your argument can't be salvaged at all, even by assuming God made the machine. If the sun rose in the east this morning, that is an accident of such unimaginably low probability that it might as well be taken as randomness. Or determinism. Your argument can't tell the difference.

tl;dr: Unimaginably low probability events happen all the time. Look around. What is the prior probability that you are you, and sitting in this particular room with its particular arrangement of stuff? What are the odds that there's any stuff at all? The (prior) odds are virtually zero. Yet here you are. Does that make the universe random, or deterministic? Your thought experiment gives no clue.

The problem with randomness is we don't have a rigorous definition of it. Also, if 100 events have an equal probability of happening they are said to be random but one of them does happen and there must be a reason for that. It may be that randomness is simply a subjective human concept.

What is the prior probability that you are you, and sitting in this particular room with its particular arrangement of stuff? What are the odds that there's any stuff at all? The odds are virtually zero. Yet here you are. — fishfry

It seems everything that happens in human life is highly unlikely - what are the chances of you and all the people in the cinema being there? But, as you say, you are all there!

Also, if 100 events have an equal probability of happening they are said to be random but one of them does happen and there must be a reason for that. — EnPassant

Lottery paradox. It's rational to conclude that you won't win, therefore you shouldn't play. But somebody must win.

https://en.wikipedia.org/wiki/Lottery_paradox

Randomness seems like a type of freedom while determinism is Newtonian. I appreciate that you saw no difference between between them in my first post. That was the purpose. If any thing can happen with randomness then it seems any result must be accepted. So my conclusion is that pure randomness cannot be understood by us (that is, randomness has parameters)

Lottery paradox. It's rational to conclude that you won't win, therefore you shouldn't play. But somebody must win.

I honestly don't see what is paradoxical about that.

It is rational to conclude that it is very unlikely for my ticket to win, but it is not rational to conclude that it is certain that it won't win.

Something may happen despite being improbable. Probability is all we need in practice.

EDIT:
At first I applied a GOF-test, but I ignored that I need more to data to justify using that test. My informal response is that there are only 10 strings as concentrated as 5555555555, and that is stronger intuitive evidence against randomness than 2343549094.

Any sequence it puts out is equally unlikely, or likely. it's just that some sequences look more unlikely than others.

That's true, and that's a good point. Each string of ten digits is equally likely, but it's strange that we got such a monotonous sample, all in the same category of '5.' What was really needed in my first post was a goodness-of-fit test, though we don't have enough data to justify. Probably a simulation approach to getting a p-value would be appropriate.

Well, if the machine were random, it could display anything from 0000000000 to 9999999999 at the designated date every year. The possibility space includes the number 5555555555 and it being the first number displayed doesn't, in any way, aid us in deciding whether the machine is a true random number generator or not. We would need a sample of numbers over a very long time to come to that conclusion because only from comparing the frequency of each digit to theoretical probability calculations can we make an inference of this sort.

Perhaps what threw you off and made you ask this question was that all the ten numbers are 5, perhaps reminding you of slot machines whose jackpot win sequence is usually a series of identical digits. This creates an illusion that from, say, a list of numbers from 000 to 999, to get numbers like 111, 222, 333, 444, you get the idea, are rarer or special or require skill or for the laws of chance to be violated i.e. we believe, erroneously, that we can't get numbers like these by chance. Wrong! getting 5555555555 or 6666666666 or 1111111111 or similar repeating digits is as likely as getting numbers like 9478321564 with non-repeating digits. The bottom line - there's nothing fishy going on with 5555555555.

Thanks everyone.

Logicism is the breaking down of mathematics into its simplest components. I don't know if this is possible with randomness, because since any outcome is as likely as another, we can't really say what the probability of rolling a 5 is, yet we know how it works in the real (i.e. practical) world

There is no test for randomness.

That's why they say god thorws dice. Nobody can generate random numbers, not because it's impossible, but because it is not possible to test for it being truly random.

That's why they say god throws dice. Nobody can generate random numbers, not because it's impossible, but because it is not possible to test for it being truly random. — god must be atheist

I'm thinking it's not the number that's random, but the process that produces it. If a series of numbers looks random, arguably it is. A test would be whether the next numbers in the series could be predicted - if the series proved to be an oracle for its own successive members.

And I can understand why computers generate pseudo-random numbers. But it seems fairly easy to imagine a physical system that would produce series of random numbers. Lottery machines seem to do it.

But it seems fairly easy to imagine a physical system that would produce series of random numbers. — tim wood

Back in my meteorology days one approach was to measure small changes in atmospheric pressure. But not a guarantee of randomness.

Back in my meteorology days one approach was to measure small changes in atmospheric pressure. But not a guarantee of randomness. — jgill

Goes to just what a random number is. I have heard one definition that a number is "random" when it is the shortest description of itself.

My own notion, a bit more complicated, is two ping-pong ball lottery machines. The ball from the first being an oracle in some way for selecting the ball from the second. Example: if the ball from the first machine is a four, then, the fourth ball from the second is selected. And so forth. I can see that one possible problem with the atmospheric pressure would be that it various continuously, although I imagine that could be resolved without too much trouble.

Btw, do you buy the idea that a sequence is random if the next cannot be guessed from the series - which probably requires actually generating the next number? .

Btw, do you buy the idea that a sequence is random if the next cannot be guessed from the series - which probably requires actually generating the next number? . — tim wood

I've been stumped by brain teasers that have you try to determine the next number in a sequence, so, no, not necessarily random. This is another math topic that not too many professionals give much thought to - unless you work in that area. If I need to generate a random sequence for a computer program I just go to that command and not worry about it. :cool:

I've been stumped by brain teasers that have you try to determine the next number in a sequence, — jgill

And they never tell you that any number you choose is correct. Of course the idea is that "they" have a rule in mind that you're supposed to divine. But if you choose your own rule, then your good!

We should keep in mind that pRNGs are a big part of modern technology and that there are lots of test. http://www-users.math.umn.edu/~garrett/students/reu/pRNGs.pdf

One can argue that no test is 'perfect,' but then we have to figure out what is meant by 'perfect.' In the same way there are some good definitions of randomness.

The possibility space includes the number 5555555555 and it being the first number displayed doesn't, in any way, aid us in deciding whether the machine is a true random number generator or not. — TheMadFool

I don't think it's that clear. While statistical hypothesis tests are never conclusive (because haunted by the possibility of type-I and type-II error, I think it's fair to say that lower p-values (if understood) are experienced as stronger evidence against the null hypothesis (in this case randomness.)

While each string of digits is equally likely, we can categorize strings by how spread out over the 10 categories they are. The more spread-out strings are more common and therefore more likely. The string is all 5s has only 9 other strings of similar extremity. That's a 1 in 10^9 chance of such a concentration, which is strong evidence against randomness.

The more spread-out strings are more common and therefore more likely. The string is all 5s has only 9 other strings of similar extremity. That's a 1 in 10^9 chance of such a concentration, which is strong evidence against randomness. — T H E

This is thinking that has been led astray, and with probabilities that happens easily. The only remedy is to be very careful about what you're looking at, and having decided that, not to be distracted into confusing yourself with problems that are just plain different.

While each string of digits is equally likely, we can categorize strings by how spread out over the 10 categories they are. The more spread-out strings are more common and therefore more likely. The string is all 5s has only 9 other strings of similar extremity. That's a 1 in 10^9 chance of such a concentration, which is strong evidence against randomness. — T H E

That's only because you're lumping all the "evenly spread" events together. There are far more of them. Whatever outcome you got was incredibly unlikely. The fact that it's a member of an arbitrarily large class of outcomes doesn't make any difference except psychologically.

Let's consider a simpler example, so you can see where I am coming from.

Let's flip a coin ten times to test it for fairness and get HHHHHHHHHH.

If H_0: p= 0.5 and H_A: p != 0.5, then the p-value is 2/2^10 = 1/2^9 ~ 0.002.

Is this evidence against the fairness of the coin? If the coin is fair, the chance of such an extreme value is about 0.2%. Let's make this concrete: if you were thinking of using the coin for practical purposes, would you trust it?

IMO, the problem at hand is more mathematically complicated (with 10 categories) but subject to the same principle.

In null hypothesis significance testing, the p-value[note 1] is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct.[2][3] A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Reporting p-values of statistical tests is common practice in academic publications of many quantitative fields. Since the precise meaning of p-value is hard to grasp, misuse is widespread and has been a major topic in metascience.[4][5]
...
Null hypothesis testing is a reductio ad absurdum argument adapted to statistics. In essence, a claim is assumed valid if its counterclaim is highly implausible.
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Thus, the only hypothesis that needs to be specified in this test and which embodies the counterclaim is referred to as the null hypothesis; that is, the hypothesis to be nullified. A result is said to be statistically significant if it allows us to reject the null hypothesis. The result, being statistically significant, was highly improbable if the null hypothesis is assumed to be true. A rejection of the null hypothesis implies that the correct hypothesis lies in the logical complement of the null hypothesis. — wiki

https://en.wikipedia.org/wiki/P-value
emphasis added

That's only because you're lumping all the "evenly spread" events together. There are far more of them. Whatever outcome you got was incredibly unlikely. The fact that it's a member of an arbitrarily large class of outcomes doesn't make any difference except psychologically. — fishfry

But what then do you make of testing the coin for fairness as in my reply to tim?

But what then do you make of testing the coin for fairness as in my reply to tim? — T H E

It's commonly accepted that the coin is fair if a long sequence of flips meets the usual tests for statistical randomness, which is your point and several other people's point. But that's also the definition of pseudorandom. We can generate completely deterministic sequences of numbers that satisfy every known statistical test for randomness. So if we see a statistically random sequence of numbers or coin flips, we actually can't conclude anything at all for certain. All we can say is that all heads doesn't look random, and a statistically random-looking sequence does look random. But we can never be sure. We can say how things appear; but we can't say for sure how things are.

I agree. I don't see any way around type-I and type-II error. This is one reason I like to frame things practically. Let's say we are doing quality control on some product. At what point do we decide to reject the object? We know that an error is always possible, but we also don't want to ignore p-values and just guess.

We can generate completely deterministic sequences of numbers that satisfy every known statistical test for randomness. — fishfry

I agree here too. As we discussed earlier, we seem to have some intuition of perfect randomness (the ideal fair coin.) And our tests seem to be built on this intuition. But as you say, we have PRNGS (which I understand to be defined in terms of being actually deterministic), that pass such tests.

we seem to have some intuition of perfect randomness — T H E

There's a recent thread on the nature of probability, a notoriously tricky philosophical subject. Perhaps there are some clues there. Also I enjoyed Nassim Taleb's book, Fooled by Randomness, in which he argues that we often confuse random events with meaningful ones, as in survivor bias.

There seems to me something infinite about randomness. Light exists in eternity says the physicist, and this is a type of infinity. Randomness seems to "break rules" to an infinite degree yet how we experience it in the real world indicates it follows certain rules

I've read much of Fooled by Randomness. Taleb is great. I like that he programs simulations (and I'm quite fascinated by PRNGs and all that they can be used for.)

There seems to me something infinite about randomness. — Gregory

If you look into algorithmic information theory, you'll see randomness as irreducible. It has no useless space in it. It's thick. And a truly random (infinite) sequence contains an infinite amount of information (which can be thought of as an unpredictable string of yes/no answers to a countably infinite number of questions.)

A test would be whether the next numbers in the series could be predicted - if the series proved to be an oracle for its own successive members. — tim wood

Any sequence of numbers can be described as a sequence of a polynomial function. Not only by one precise, exact and fitting polynomial function, but actually an infinite number of them.

So the next successive number can always be predicted. Or else explained.

This test you propose is not the one that's going to work in establishing randomness of a sequence of numbers.

This is all very confusing. I just I started a tar puzzle. If rolling a dice has a probability of coming up with a 6 only 1 out of six times, each time I roll the dice the probability is the same. Yet there is no equivalence between each roll taken by itself and the sun total

I guess nothing is completely random. When we see things floating in space, they move randomnly. But they still make sense; they are not irrational in their movements