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Comment by fainpul | Hacker News Reader

> So if you only shoot 100/100 with your coin, that's definitely weird.

Not if you only try once.

Even if you shoot only once, you still have a higher chance of hitting something slightly off the middle than the perfect 100/100. And this because that's one point-precise result (100/100) vs. a cumulated range of individually less-probable results, but more probable when taken as a whole.

For a fair coin, hitting 100/100 is ~5%, vs. ~30% falling in [97; 103] \ {100}. You can simulate here: https://www.omnicalculator.com/statistics/coin-flip-probabil...

> you still have a higher chance of hitting something slightly off the middle than the perfect 100/100

That's because "something slightly off the middle" is a large group of possible results. Of course you can assemble a group of possible results that has a higher likelihood than a single result (even the most likely single result!). But you could make the same argument for any single result, including one of the results in your "slightly off the middle" group. Did you get 97 heads? Well you'd have a higher likelihood of getting between 98 and 103 heads. In fact, for any result you get, it would have been more likely to get some other result! :D

> But you could make the same argument for any single result

Isn't that the point? The odds of getting the "most likely result" are lower than the odds of getting not the most likely result. Therefore, getting exactly 100/100 heads and tails would be unlikely!

But as I said, getting any one specific result is less likely than getting another other possible result. And the disparity in likelihoods is greater for any one specific result other than the 50% split.

I think the disagreement is about what that unlikeliness implies. "Aha! You got any result? Clearly you're lying!"... I'm not sure how far that gets you.

There's probably a dorm-quality insight there about the supreme unlikeliness of being, though: out of all the possible universes, this one, etc...

Let's look at the original quote:

> "Remember, if you flip a coin 200 times and it comes heads up exactly 100 times, the chances are the coin is actually unfair. You should expect to see something like 93 or 107 instead".

Inverting the statement makes it read something like this:

You are more likely to not get 100/100 than you are to get exactly 100/100

...which is exactly what I was saying. Nobody is arguing that there is a single value that might be more likely than 100/100. Rather, the argument is that a 100/100 result is suspiciously fair.

Try thinking of it this way: you're in highschool and your stats teacher gives you a homework assignment to flip a coin 200 times. You respect her and don't want to disappoint her, but at the same time the assignment is pointlessly tedious and you want to write down a fake result which will convince her you actually did it.

A slightly imperfect split is more likely to convince your teacher that you did the assignment. Intuitively this should be obvious.

Should that be 25% for 97..193 excluding 100?

“[97; 103] \ {100}” means the interval [97; 103] without the set {100}; so no, still ~30%.

I'm sorry, but try what once? 200 flips once?

In statistics, various examples (e.g., coin flips) often stand in for other activities which might prove expensive or infeasible to make repeated tries of.

For "coin flips", read: human lives, financial investments, scientific observations, historical observations (how many distinct historical analogues are available to you), dating (see, e.g., "the secretary problem" or similar optimal stopping / search bounding problems).

With sufficiently low-numbered trial phenomena, statistics gets weird. A classic example would be the anthropic principle: how is it that the Universe is so perfectly suited for human beings, a life-form which can contemplate why the Universe it so perfectly suited for it? Well, if the Universe were not so suited ... we wouldn't be here to ponder that question. The US judge Richard Posner made a similar observation in his book "Catastrophe: Risk and Response" tackles the common objection to doomsday predictions that all have so far proved false. But then, of all the worlds in which a mass extinction event has wiped out all life prior to the emergence of a technologically-advanced species, there would be no (indegenous) witnesses to the fact. We are only here to ponder that question because utter annihilation did not occur. As Posner writes:

By definition, all but the last doomsday prediction is false. Yet it does not follow, as many seem to think, that all doomsday predictions must be false; what follow is only that all such predictions but one are false.

-Richard A. Posner, Catastrophe: Risk and Response, p. 13.

<https://archive.org/details/catastropheriskr00posn/page/13/m...>

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