No, it's definitely some evidence. It's not a certainty or anything, it's just SOME evidence. I stand by everything I said before, and repeat my question: what's your prior? This post seems to imply that your prior is that all coin-flips must be exactly 50-50... this is pretty short-sighted.
Dude what? I don't know what "what's your prior" means but it sure seems moot. I could have said "there are thirteen people left" or whatever to make the scenario seem more random; it seemed pointless. Why you would think that I would think that coins are perfect 50-50 machines is a mystery. I guess you just didn't like that I tied my entertaining scenario, which I was considering posting anyway, into your statement, and so you thought, let's call him something?
Maybe you just have a short-sighted way of using the word "evidence." We have motive here - who doesn't want to win a coin-flipping contest? We have suspects. We have no evidence. Nothing noteworthy has happened. It only looks noteworthy when you take it out of context. To get evidence, we would need to try more flips, or study tapes of the people flipping, or analyze the coins, etc., none of which have happened yet. We do not have that evidence.
I have people flipping different coins, because that's my fun scenario for showcasing the point I actually wanted to make, which was nothing to do with this; and so, you can attack that, try to say that there is some small amount of evidence of foul play simply because there was no guarantee that even a single coin would flip ten heads, no matter how likely it was that several coins would. Okay so. I can counter your initial assertion trivially by flipping a single coin enough times. Ten heads in a row is no evidence of bias inside the context of zillions of flips that did not show bias, which could well be the scenario Synthesizer was describing ("You are flipping a coin [zillions of times]..."). At a certain point it would be suspicious if it *didn't* get ten heads in a row sometimes. I don't know how you rig a coin like that, but you know, we could be living in a simulation or something.
You start your post with "I can arrange for it not to be!"
This is wrong, which is fine - man, people are wrong all the time. The tone created therein of "Gee-bet-you-hadn't-thought-of-THAT, I am much cleverer than you are, here's a nice example to show it" is somewhat irksome, but again, not THAT bad. Combining them is not so hot.
But let's dig into the particulars of the situation.
Yes, long series flips of the same side on a coin that has a mythical "truly 50%" head-tail rate on any given Bernoulli trial do in fact happen with a particular probability - namely, (.5)^N, or if it could be either side and you aren't specifying that they need to all actually be heads, then there are two ways of doing this, and it turns into (.5)^(N-1), where N in either case is the number of consecutive flips. Yes, this can indeed happen.
My claim is that 10 flips (even one flip for that matter) constitutes
some evidence that it isn't
fair. By fair here, I mean the mythical 50-50 coin. I am
not claiming that this is enough evidence to make me think that an unfair coin is most probable, or even very likely at all. All I am saying is that after witnessing 10 consecutive heads flips, you should now think it more likely that the coin isn't 'fair' than before you witnessed the 10 consecutive head flips. I am not saying how much more likely. Just more likely. This is what I mean by evidence. The only ways to escape this are to reject probability axioms or to set yourself as absolutely undeniably certain on a fair coin before you start things. My short-sighted comment pertains to the latter stance (and if you note, it's a claim about the position, not the person who holds it), since the former stance seems incompatible to me with holding that there is a probability of the coin, much less that it's 50%.
Again, whether or not it is 'noteworthy' is a matter of opinion, but I clearly think it is worth making a note of - this is why I did.
Your counters fail to grasp the basic point I am trying to make - in fact, they are an illustration of the particular fallacy which I'm trying to point out. That improbable things sometimes will happen DOES NOT imply that more information shouldn't revise initial estimates. This is what the "What's your prior" comment is getting at. I am asking you, before flipping the coin a single time, what did you believe the distribution of 'true percentage' of time that the coin was flipped would come up heads was? Now, it seems that your entire distribution of probabilities is 50%, which is, I am arguing, not a good choice. Because there is some probability of a trick coin, or just a slightly imbalanced one, or trick throws, or who knows what. It might be a REALLY REALLY small chance, but you ought not to say that it's just zero. How big this non-zero chance is is going to determine how many heads in a row you will take as sufficient evidence that it's not 50-50. So if you think there's a one-in-a-million chance that it's double-headed, you have pretty good evidence after about 20 flips of all heads that this is starting to look better than a 50-50 guess.
In your actual scenario, things are much more stilted. First of all, studies have shown a long-term average of like 51% that the coin will come up the same way it started with. This is somewhat dependent on the coin, but it's much more dependent on the flip. Some people flip in a way that comes up the same quite a lot (conversely some people have it switch a lot). Further, in a contest like this, I expect to see a much much higher percentage of double-headed coins than I would just randomly picking a coin up off the street - people do weird things in contests. Beyond this, people are more likely to TRY to do a trick toss or have a rigged coin or something in a contest, because yeah, we do want to win. So, after 10 throws, I am not going to guess that it's double-headed. But I will update my original estimate (which probably would have been 51-52%) up a ways, probably ending up somewhere in the 55-60% range. My guess would be that they are significantly more likely than a 'fair coin' to flip heads next, probably because they have a flip that tends to rotate the coin an even number of times. Now, if I actually had more specific information on this stuff, then I could do better than that. But if you actually run this contest with enough people (the more the better), I would guess that the long-term average of these 'survivors' will be statistically significantly higher than 50% heads. And this is based on the evidence of their first 10 flips.
