If you cherry-pick that statement in isolation you're technically correct. But in practice the whole comment you're replying to is actually correct, in this case if you saw the number four, one would be correct in concluding that's unlikely and searching other low digit numbers, the proof of that is that it happened here.

I think they are right and you are wrong. If I understand properly you say:

"Playing roulette, in isolation after RED - RED - RED - RED - RED, there is not a higher chance of getting BLACK than RED in the next run. But in practice, one would be correct in concluding that it will probably be BLACK. The proof is that in our test, the next run gave BLACK".

Said differently, it was not particularly unlikely that the number would be 4. But maybe it got the attention of the person who decided to try assuming that it was a mistake, and they were indeed lucky.

I think it's more like seeing RED - RED - RED - RED - RED and concluding the roulette is broken

My understanding is more that it's like seeing RED - RED - RED - RED - RED and wondering if the roulette is broken, testing it, and realising it is.

The hypothesis is "I see a small number, I can test if it is broken or not... oh it is". Could totally have been "no it's all good, just happened to be a small number".

They had multiple keys and 7 of the keys were numbers less than 256.

Yes, and that is what's statistically impossible. They said one cert.

> In this case one cert gets defeated and you go the odds of that number being that low are 2^-384.

There are several very practical reasons (and not emotional) reasons to expect that "4" is the result of bad implementation.

I can think any number of programming mistakes that will result in a supposed (pseudo-)random number generator to be be biased to very small numbers. I have made such mistakes myself.

By any reasonable estimate, the probability of randomly picking a number below 256 is MUCH smaller than the probability of someone having made a mistake.

So you say that programming mistakes make small numbers more likely to be picked, and therefore seeing a small number is less likely than seeing a bigger number?

That doesn't make sense, does it?

The claim is “if I generate a random number with equal distribution in a gigantic range and get a very tiny number, it’s more likely that there’s a mistake somewhere than that I just got this number randomly”.

Which is... wrong?

If you generate a random number with equal distribution in a gigantic range, 0 is just as likely to happen as any other number in the range, right?

That's the meaning of "equal distribution".

I think you misunderstood the point.

In a working RNG, every number has equal probability.

But there's a probability <1 that the RNG is working as it should and a probability >0 that the RNG erroneously generates numbers that aren't equally distributed.

And because the range of possible numbers is so large and the probability of a mistake is not that tiny, a small result is more likely to be an error than a random result that happens to be small.

If I write a program to flip a coin and it generates heads 30 times in a row, is it more likely that this is just a random result or that I made a mistake in my programming?

One probability is 2^30, and the probability that someone makes an error resulting in this pattern is probably higher than 1/2^30.

Right, ok I get it now!

Huh? Smaller numbers are a smaller part of the space.

So if I take a random number between 1 and 100, you think it's less likely that I pick 4 than, say, 83?

No but it's less likely to pick a number <10 than >50.

So what? It's more likely to pick a number <10 than a number between 42 and 44.

I don't see why you would get downvoted for that. You're right.

It just happened that that particular number got the attention of a human, which was luck. But in terms of statistics, this number was not less likely than another one.