What I'm trying to say: if you get 100 / 100, that's not a sign of an unfair coin, it's the strongest sign for a fair coin you can get.
But the summed probability of the “not too far away results” is much higher, i.e. P([93, 107]\{100}) > P([100]).
So if you only shoot 100/100 with your coin, that's definitely weird.
You could just as easily say "you should be suspicious if you flip a coin 200 times and get exactly 93 heads, because it's far more likely to get between 99 and 187 heads."
For example, you take the top five cards, and you get a royal flush of diamonds in ascending order. In theory, this sequence is no more or less probable than any other sequence being taken from a randomly shuffled deck. But given that this sequence has special significance to people, there's a very good reason to think that this indicates that the deck is not randomly shuffled.
In theory terms, you can't just look at the probability of getting this result from a fair coin (or deck or whatever). You have to look at that probability, and the probability that the coin (deck etc.) is biased, and that a biased coin would produce the outcome you got.
If you flip a coin that feels and appears perfectly ordinary and you get exactly 100 heads and 100 tails, you should still be pretty confident that it's unbiased. If you ask somebody else to flip a coin 200 times, and you can't actually see them, and you know they're lazy, and they come back and report exactly 100/100, that's a good indicator they didn't do the flips.
Eh, this only makes sense if you're incorporating information about who set up the experiment in your statistical model. If you somehow knew that there's a 50% probability that you were given a fair coin and a 50% probability that you were given an unfair coin that lands on the opposite side of its previous flip 90% of the time, then yes, you could incorporate this sort of knowledge into your analysis of your single trial of 200 flips.
Not if you only try once.
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...
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
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!
There's probably a dorm-quality insight there about the supreme unlikeliness of being, though: out of all the possible universes, this one, etc...
A slightly imperfect split is more likely to convince your teacher that you did the assignment. Intuitively this should be obvious.
> "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.
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...>
That's true of every result. If you're using this to conclude you have a weird coin then every coin is weird.
93 heads, 107 tails and
93 tails, 107 heads
but not between
100 heads, 100 tails and
100 tails, 100 heads.
So while those are two results (93/107 and 107/93), they really only count as two separate outcomes if you pre-specify that the first number is heads.
If instead you consider symmetries, where there are 2 ways to get 93/107 and only one wall to get 100/100, then you have more likelihood for the 93/107 outcome because you have two ways to get it.
It will be a bell curve with 100/100 as the peak
Intuitively, this seems right to me, but sometime statistics do not follow intuition.
"More likely"? How can you even estimate the likelihood of the coin being "an imperfect PRNG" based on a single trial of 200 flips?
Graphing here https://www.wolframalpha.com/input?i=graph+%28%28N%2F2+choos... and it does look like it's only a slight improvement in likelihood, so I did overstate the claim. A more interesting case would be to look at some self-correcting physical process.
If a student was tasked with determining some physical constant with an experiment and they got it exactly right to 20th decimal place - I'll check their data twice or thrice. Just saying. You continue believing it was the most likely value ;)