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I suspect many of the “bad” responses are smart aleks saying “11111111” is just as likely a “62536164”.

Is 11111111 less likely than another sequence?

It’s just as likely (or unlikely) as any particular sequence.

Precisely. A difference only arises when comparing sets of strings. In our example, the number of elements in the set uniform strings aaaa...a of length L is equal to the number of distinct symbols N, while the number of district strings of length L is N^L. So if you ask "will the string be uniform?" (which is the only reason you thought up aaaa..a in the first place) you find the probability is exponentially small for long strings: N^(1-L). Really if you tend to choose a string for any particular reason other than chance, it can be exploited as long as the reason can be guessed (e.g. the dice rolls 31415 write out the digits of pi.).

The fact it’s equally likely as any other sequence means it should be very unlikely to appear in the study’s 3429 samples, and extremely unlikely to show up more than once.

You can estimate the probability distribution that generates this sequence as 0: 0, 1: 1. This is as far from 0: 0.5, 1: 0.5 (a fair coin toss) as you can get.

Comparing mean and std dev can be used to estimate the distance between two distributions. See also, statistical testing.

Depends.

If you know it has been generated by a valid random number generator, then no.

But if you know there is a chance it came from something other than a valid random number generator, then you would have to classify sequences like 11111111 in a "highly suspicious" category.

This is the wrong question. Think instead about whether a sequence like 1111111 is more likely to be produced by a fair die or a loaded one.

Helps to think of it in binary, 11111111 (bin) == 255 (dec). Also helps to define the space of possible outcomes, 00000000 -> 11111111. Then ask, is this a discrete event or a sequence of discrete events? I.e. if we have a roulette wheel with 256 slots, and throw a ball in, then the chance of the ball falling in any slot is 1/256.

But what if we say, we're going to generate that binary sequence by 8 successive flips of a coin, and we are aiming for 11111111 specifically? Then we have to multiply eight times. (0.5)^8 == .00390625 == 1/256

Where it gets a bit tricky is if we ask people to place bets after each successive flip of the coin. For example, starting with no flips, ask players to bet on the likelihood of 8 heads in a row. Next round, bet on the likelihood of 7 heads in a row, knowing the first was a head, etc. What minimal odds should the house give after each flip in order to reliably turn a profit on this game? Does it matter how many players are at the table when it comes to calculating those odds?

It’s about the reference class. The class of allsame sequences has only a few members. The class of … random numbers all over ala 71833791 has a lot of members. So the probability of seeing the former class is tiny.

Think of it in terms of coin tosses. Two heads is less likely than a head and a tail, because there are two sequences, 10, and 01, that map to the class of having one tail and one head. But there is only a single sequence of 00 in the class of all heads.

111111 is just as likely as any other number. However, in practice, humans are far more likely to think of 111111 than other numbers, so we exploit the difference between the probability a human guesses a number vs. the probability of rolling a number on a fair dice. "Did I just roll a sequence X or am I lying?" vs. "What is the probability I roll sequence X next?" are quite different questions, if by "lying" you mean you are not drawing the sequence from a fair, random source.

111111 is just as likely as any other *SEQUENCE* of numbers. This is a little confusing because the "sequential" requirement is somewhat masked by the repeating sequence used by the example. however, the odds of rolling six 1's is 0.00002143347.

considering the non-sequential set of rolls 625631 you have the odds of exactly two 6's at 0.201 and also one 2 5 3 and 1 each at 0.402

0.402^4 * 0.201 = 0.00524928641, or ~244 x more likely.

There are many ways of slicing and dicing things so that you group different sequences, e.g. group permutations as the same as you have done. But by "number" I was referring to a sequence of symbols drawn from a uniform distribution, so order matters. (On the website, you also choose options sequentially, anyways.)

No, it is not. On average you expect equal number of 1s and 0s. All 1s won't be "equally as likely" .

All 1s isn't equally as likely as the entire class of outcomes that are not all 1s, but you can also say that about every other outcome (as long as we're talking about distinguishable dice or a sequence.)

You are really confused about this. In general a randomly generated sequence on average will have equal 1s and 0s. Try generating 100K sequences and then count the sequences with:

1) equal 1s and 0s

2) all 1s

3) all 0s

See which one is more likely.

You missed the parent’s point. “Equal 1s and 0s” is not a sequence; it’s a class of sequences. So the fact that sequences with that property are more common than the specific sequence “all 1s” is true but doesn’t answer the question. 11110000 is a different sequence than 10101010. The question this thread is exploring is whether 1111111 is more likely than any other sequence, e.g. than 1111000 in particular (or insert any other sequence). And of course the answer is no.

I believe what pessimizer means is that while "all 1s" is not as likely as "equal 1s and 0s", it is just as likely as any individual string of 1s and 0s - for example, 111111 is just as likely as 100110.

The point the parent is making is that 1) represents a class of results rather than a single result, and any single member of that class is equally as likely as 2) or 3). Obviously the class as a whole is more likely than any other single result, but that's a different assertion.

> we only learned that older people pick all-H or all-T more.

I think we learn that older people are less willing to put up with silly instructions.

That’s my interpretation too, fwiw. But I stuck with the narrower facts

Why doesn't entropy or kolmigorov complexity qualify as a valid metric here? A random process will produce a highly entropic result or something with high kolmogorov complexity, whereas something biased will produce something with less (like HHHHHHHHH).

Edit: saw this other comment ( https://news.ycombinator.com/item?id=31224738 ) about how kolmogorov complexity is a bad metric and am now second guessing myself

“Sequence looks random” is not nonsensical. The authors say your sequence should be indistinguishable from e.g. 12 coin tosses. This would have a uniform distribution.

One approach would be estimate the probability distribution from the input sequence and calculate the KL-divergence [1] of that to the uniform distribution. This gives one objective measure of randomness. There are many others!

TL;DR: There are definitions of randomness that can be tested against.

[1] https://en.m.wikipedia.org/wiki/Kullback–Leibler_divergence

Exactly, it's impotant to chose a definition of randomness first.

The test specifies the random distribution very clearly in each case, though.

You can define terms however you want, and thus devise any measure you want. But that’s not a definition of randomness I recognize. Perhaps entropy. Randomness is not a property of the result; it’s a property of the process used to generate it.

Let’s imagine applying your measure to a whole bunch of sequences, each of length N, with the elements of each actually drawn from a uniform distribution. You measure the randomness of each one. You’ll get a range of DLK results, distributed from, in your interpretation, “very random” to “not that random”. All-H or mostly-H will come up sometimes, the distribution estimator will return a skewed result, and it will get a divergent “score”. But everything came from the same distribution. So we’re now measuring the output of an actually random process and saying “we’ll, it’s usually random but not quite always”

In contrast, let’s try your method on a different population of sequences, where instead of pulling the sequence elements from a distribution, every sequence is a hardcoded copy of HTHTHT… That gives “perfectly random”, even though it was very far from a random process.

That’s close to what the study authors are doing here, except with a different definition of “random looking”. It’s measuring a property of a sequence, but it isn’t whether it was randomly generated.

We could debate about whether this is a good measure of “random looking” and there could be lots of alternatives with no objectively best. But that is my point: if I ask “make me something random-looking”, I am only asking “how closely does your measure of post-facto randomness match mine?” An actual random sequence would be, well, random.

Agreed. This ends up being a test of one of a few things:

1. Whether people of different age groups agree that random sequences are more complex as defined by the study’s notion of complexity.

2. Whether people of groups have the physical ability to generate those sequences in a web browser.

3. Whether people of groups have the mental ability to generate those sequences.

Of these, the study purports to measure 3, but it’s actually conflating all of these.

> “Make a sequence that looks random” is sort of a nonsensical ask. Looks random to whom?

To other people. They specified this in the study instructions. (Not defending the study, I think it's flawed too.)

But they didn’t measure that. They’d have to measure that by showing a bunch of people each sequence and asking “is this random?” Instead, they threw a complexity formula at it, which measures some specific thing, but not “other people”

Though perhaps there is a body of existing literature showing that their complexity estimator matches people’s assessment of “randomness”? If so, does it include people over 60?

> But they didn’t measure that.

That's correct.

> They’d have to measure that by showing a bunch of people each sequence and asking “is this random?”

Agreed!

> Instead, they threw a complexity formula at it, which measures some specific thing, but not “other people”

Yep, they botched the study.

Nonetheless, the "ask" that you criticized was fine. It explained that the subject should craft a pattern which would appear random to other people, and imo there's nothing wrong with that ask. The mistakes came afterwards.

Alright, fair enough

I agree with this.

I think 1 1 2 3 4 looks more real than 1 2 3 4 5

In that a draw of 5 numbers is likely to include at least one number twice and one number zero times.

But is that the goal? Is it communicable to the audience

Agreed, I deliberately used the linux RNG, and it generated T T T T T T T H H H H H

Probably why I got an "age" of 60.

The age predictor seems broken, my sequence was scored as very random (90th percentile) and it still picked 60 for me.

Ironically based on eyeballing the chart it actually did predict my age within 5 years