Note that "odds", or more formally, probability, is a mathematical concept and therefore obeys mathematical laws rather than natural ones. The mathematical concept of probability does not control what happens to a tossed coin, rather it is a model for predicting the result of the toss (or a model for discussing why it is not predictable, as the case may be). It is conceivable that the "real world" does not behave in the way modeled by mathematics and that gamblers fallacy is not a fallacy at all. However, the math is close enough that real world deviation is at most very small.

Thanks, that gets much closer to my concerns. Though I still have no good prospect of sorting these "concepts" out, so I veer towards pragmatism.

See also: math.stackexchange.com/questions/1125087/…

Why the close vote? this is an awesome Q.

On the contrary, after 10 heads in a row I would suspect the coin to be not a fair but a biased coin and would set my money on heads.

Related: stats.stackexchange.com/questions/136870/…

I'd suggest the question is correctly tagged as 'philosophy-of-maths'; but agreed there is a tendency to answer questions like this in a math manner, rather than philosophical one.

You should read the first scene of Rosencrantz and Guildenstern Are Dead by Tom Stoppard. "A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability."

@CamilStaps. I agree that there are purely mathematical answers, but I was hoping for more... a better insight into the whys and wherefores of probability in relation to the physical world, induction, and common sense. For me at least "gambler's fallacy" was clearest case where "history" doesn't count. I am not sure what sorts of assumptions go into this.

The fact that a fair coin's flips converge to 50:50 means that given enough flips the ratio converges, regardless of how uneven the ratio starts out as. Even if you flip a fair coin 99 times and through a miraculous chance you get 99 heads, once you've flipped it a million times those 99 "anomalities" don't matter anymore. Assuming the rest of the flips are about even, 500050:499950 is very close to 50:50 (and it gets even closer after more flips). That's what the convergence means: the flips don't automatically stabilize quickly, but the uneven start becomes meaningless given enough flips.

I'm not sure if this was addressed in the other answers: say I flip a coin once and it lands heads. Now say I think about what I expect the ratio of heads to tails to look like in the long run. I expect all future coin tosses to be 50/50 between heads and tails. That does not change. And therefore I expect the ratio to converge to one. But convergence does not imply equivalence. I would expect, after 2000 more tosses, to have 1001 heads to 1000 tails. After 2001, I'd be 50/50 between 1001:1001 and 1002:1000. But these are still closer to 50/50 than my initial 1/0. Convergence != equivalence.

This question deserves an answer discussing Brownian motion.

@BenVoigt. Please provide if you wish. I have also posted a related question on the "physical" aspects of the coin toss.

Nelson, what you're referring to is the law of large numbers. The issue you seem to be having troubles with is that the ratio heads/tails converges to 1, while the expected difference |heads-tails| grows unbounded. These are not the same thing.

@PieterB That reminds me of a quote I saw on the door of one of the professors in my university's math department: Many people think that after rolling 1000 heads in a row, the odds of the next toss being heads is very small. A mathematician knows that the probability is still 50%. Any fool knows it's a two-headed coin.

Probability is actually very simple and intuitive - people just avoid calculating it. Let's stay with coins. You get the probability of your result by counting the ways it can be true, and the ways it can be false. If you want a series, say, TTT, there is one possible true result (TTT), and seven false results (HHH, HHT, ...) - so the odds are 1:8. When you already have TTT, and you ask for TTTT, there is one possible true result (TTTT) and one possible false result (TTTH) - so the odds are 1:1. It's that simple - you only really need to be able to count.

Thanks, but I wouldn't say it's all so intuitive. Bayesian remains controversial and Monty Hall Problem fools many mathematicians.

@ToddWilcox: Thank you for pointing out the distinction between math and physics. The fact that math uses models, not natural laws, is too often lost these days.

While an interesting question, it seems a "Simple Matter of Probability".

@JamesKingsbery. Yes, most responders have opted to do the math. But the epistemological status of probability is not entirely uncontroversial. Far from it. I still do not think I have an entirely good answer as to the "context" of having 9 heads and yet 10 heads is entirely the same probability. Under what context? What "framing" gets us from 9 heads back to the convergence? I am a bit frustrated because everyone is just repeating the textbook math. Probabilities entail assumptions a "point of view" that is not obvious.

Well, that's the thing - there is no context. Note that in bayesian statistics, you would tend to update your probability expectation if you started getting consistent patterns (e.g. 60% heads). However, that's no longer a fair toss - your question doesn't allow for loaded coins, "weird" coin tosses etc. If the probability changed with history, POV and context would be important. They aren't - historical results might adjust your expectations, but they do not change the intrinsic probability (e.g. "you guesstimated the probability wrong").

The sad thing is, while being extremely simple (the "intuitive" approach is much more complex - I mean, you just added history; that's a big deal), even educated people get this wrong. This gets outright scary when a doctor can't tell you the probability of e.g. having cancer, because he can't count the probabilities correctly. Now, the obvious answer to epistemology is "it works". You can try it yourself easily. There was even a Mythbusters episode on the Monty Hall problem, with fairly conclusive results. But it's still the simplest path that gives you the correct (observed) result.

You just tossed the coin once and it came up heads. But that coin could have been tossed thousands of times already in its "life". If it is a fair coin, this one head maybe added to the 9,999 heads and 9,999 tails it already did in the past. Now does it seem so unlikely that the next toss is 50:50 heads or tails? Also note it does not matter if the coin has been tossed that many times - we don't know if it has or not, but we can imagine that it was. These past coin tosses do not affect the next one any more than possible future tosses do.

is this a "fair" coin in that it is not physically biased to one side or the other in any physical metric? Weight, size, etc.?

In addition to all the good answers and comments here, I think what gets (some of) us in this is our tendency to look for and recognize certain patterns but not others. Tails-tails-tails-tails-tails-tails as the outcome of six coin tosses with an unbiased coin is no more and no less likely than the outcome Tails-Heads-Heads-Tails-Tails-Heads, or any other specific sequence.

One thing to keep in mind is that before you start flipping coins, your expected result is as many heads as tails. After you flip the first coin and get heads, your new expected result is one more heads than tails. And so on.

This is a deep question of epistemology - starting with "who told you the coin was fair?" I think this point is at the heart of the Frequentist Vs Baysean debate (xkcd.com/1132). You get into a tautological loop very quickly if you say that you know a coin isn't fair if the ratio isn't 50:50 - and what of a coin that always lands on alternative sides? Clearly the ratio sticks as close as possible to 50:50 but there's something very fishy about such a coin.

Assuming a fair coin, flipping 10 heads in a row is a pretty improbable event. Flipping 11 heads in a row is likely an even more improbable event. However, as the coin itself does not track how many times it was flipped and what the results of those flips were, it has no effect on the next flip. In the absence of an outside force that tracks the results of the flips, each coin flip is its own 'instance' of flipping a coin, thus each flip has an unchanged 50:50 chance of being heads or tails.

@NelsonAlexander "But the epistemological status of probability is not entirely uncontroversial." Er....are you sure about this? If you're looking for an epistemological defense of probability in general, that sounds like a different (and probably overly-broad) question.

Well, yes I am sort of thinking about an "epistemological defense of probability in general"... and I agree that's pretty broad. Hence my more specific question. There are debates on the nature of probability and I have yet to read that literature, so I'm not entirely sure how to best frame the questions. But these answers may help. I am beginning to get a sense of contexts in which information is retained and those in which it is not and what that means.

@NelsonAlexander If you're thinking of the frequentist vs Bayesian "debate," there's actually not any doubt (mathematically speaking) about Bayes' Law, and as far as I know there's no real doubt about the epistemology of frequentist probability, either.

Particle-Wave Duality...

As far as I can tell, this SEP article on "interpretations of probability" discusses the question(s) at issue. I'm not sure why this is being closed as off-topic. Maybe "unclear what you're asking", but definitely seems on topic.

@Dennis Looks like it didn't stay closed, but in any case, I think the problem is that it's not all that clear from the question that OP is looking for a defense of probability rather than simply an explanation of the mathematics in question.