So given there have been 9 heads in a row, maybe your average bear would think its more likely to come up heads than it’s genuine expected value, so I would argue that the market would probably overvalue the true likelihood (which according to you is 10/11ths), and so they would bet on heads and you would want to be short (bet on tails) if their expectation/price is greater than 10/11ths.
That was the point of the coin flip example. It was to point out that the market is not random even if it appears to be about as random as a coin flip. Information from the previous flip factors into the next flip, reducing the likelihood that any given trend will continue.
What I think you are missing is the fact that everybody knows that the way to take advantage of an inflated price is to sell short - it is not a unique insight on your part. Since it's common knowledge, obviously everybody knows that everybody else knows that the way to exploit this situation is to sell short. Therefore there is a very high probability that a significant portion of the market will bet on tails to capture the edge. This destroys your likelihood of successfully beating the edge, because too many people are going to attempt the same thing you are. Those who recognize this (and there are many) also know that the next level of exploitation is to bet on heads.
The end result is a wash - there is a 50/50 chance that the 11'th flip will be heads, not a 10/11 chance like traditional probability suggests, because everybody is trying to out-exploit everybody else. The market is anti-inductive.
Think of poker. Someone who knows the probabilities of poker hands can win in far more situations than someone who does not. They will know when to bet and when to fold, maximizing their success. However, when playing with players who know the probabilities of poker hands this strategy becomes much less successful. Because everyone is only playing cards they can win with, it is only the really lucky players who get more good cards than bad who come out ahead.
However, by exploiting the likelihood of given cards, they can pretend they have cards they do not have and convince the rest of the players to give up. This makes their probability of winning skyrocket, and the net result is that the best hand winds far less often than probability suggests. This is the result of everybody knowing how to exploit the probabilities - everybody knows to go for the edge.
In high level poker, however, the best hand wins 99% of the time. Each particular hand wins at almost exactly the rate its likelihood of appearing suggests. Why? Because everybody at the table knows the probabilities for a given hand, and everybody is going to attempt to exploit those probabilities. Furthermore, everybody knows that everybody knows this, so it is only on extremely rare occasions that someone is actually able to exploit the probabilities and win with a weaker hand. In this scenario it is extremely difficult to inflate the value of the cards you are holding by bluffing, because the person you are bluffing knows the likelihood that you have the hand you are pretending to have and can compare that to their own hand to get their chances of winning. It still works on occasion though, and can be pretty spectacular.
This is the efficiency of the market. It is anti-inductive because information flows freely. As soon as an exploit is discovered it is nullified by the fact that it cannot be hidden, and everybody will therefore take advantage of it.
In high level poker, however, the best hand wins 99% of the time.
Really? This seems surprisingly high.
I suspect there's a Pons Asinorum of probability between the bettor who thinks that you make money on horse races by betting on the horse you think will win, and the bettor who realizes that you can only make money on horse races if you find horses whose odds seem poorly calibrated relative to superior probabilistic guesses.
There is, I think, a second Pons Asinorum associated with more advanced finance, and it is the concept that markets are an anti-inductive environment.
Let's say you see me flipping a coin. It is not necessarily a fair coin. It's a biased coin, and you don't know the bias. I flip the coin nine times, and the coin comes up "heads" each time. I flip the coin a tenth time. What is the probability that it comes up heads?
If you answered "ten-elevenths, by Laplace's Rule of Succession", you are a fine scientist in ordinary environments, but you will lose money in finance.
In finance the correct reply is, "Well... if everyone else also saw the coin coming up heads... then by now the odds are probably back to fifty-fifty."
Recently on Hacker News I saw a commenter insisting that stock prices had nowhere to go but down, because the economy was in such awful shape. If stock prices have nowhere to go but down, and everyone knows it, then trades won't clear - remember, for every seller there must be a buyer - until prices have gone down far enough that there is once again a possibility of prices going up.
So you can see the bizarreness of someone saying, "Real estate prices have gone up by 10% a year for the last N years, and we've never seen a drop." This treats the market like it was the mass of an electron or something. Markets are anti-inductive. If, historically, real estate prices have always gone up, they will keep rising until they can go down.
To get an excess return - a return that pays premium interest over the going rate for that level of riskiness - you need to know something that other market participants don't, or they will rush in and bid up whatever you're buying (or bid down whatever you're selling) until the returns match prevailing rates.
If the economy is awful and everyone knows it, no one's going to buy at a price that doesn't take into account that knowledge.
If there's an obvious possibility of prices dropping further, then the market must also believe there's a probability of prices rising to make up for it, or the trades won't clear.
This elementary point has all sorts of caveats I'm not bothering to include here, like the fact that "up" and "down" is relative to the risk-free interest rate and so on. Nobody believes the market is really "efficient", and recent events suggest it is less efficient than previously believed, and I have a certain friend who says it's even less efficient than that... but still, the market does not leave hundred-dollar-bills on the table if everyone believes in them.
There was a time when the Dow systematically tended to drop on Friday and rise on Monday, and once this was noticed and published, the effect went away.
Past history, e.g. "real estate prices have always gone up", is not private info.
And the same also goes for more complicated regularities. Let's say two stock prices are historically anticorrelated - the variance in their returns moves in opposite directions. As soon as everyone believes this, hedge-fund managers will leverage up and buy both stocks. Everyone will do this, meaning that both stocks will rise. As the stocks rise, their returns get more expensive. The hedge-fund managers book profits, though, because their stocks are rising. Eventually the stock prices rise to the point they can go down. Once they do, hedge-fund managers who got in late will have to liquidate some of their assets to cover margin calls. This means that both stock prices will go down - at the same time, even though they were originally anticorrelated. Other hedge funds may lose money on the same two stocks and also sell or liquidate, driving the price down further, etcetera. The correlative structure behaves anti-inductively, because other people can observe it too.
If mortage defaults are historically uncorrelated, so that you can get an excess return on risk by buying lots of mortages and pooling them together, then people will rush in and buy lots of mortgages until (a) rates on mortgages are bid down (b) individual mortgage failure rates rise (c) mortgage failure rates become more correlated, possibly looking uncorrelated in the short-term but having more future scenarios where they all fail at once.
Whatever is believed in, stops being real. The market is literally anti-inductive rather than anti-regular - it's the regularity that enough participants induce, which therefore goes away.
This, as I understand it, is the standard theory of "efficient markets", which should perhaps have been called "inexploitable markets" or "markets that are not easy to exploit because others are already trying to exploit them". Should I have made a mistake thereof, let me be corrected.
Now it's not surprising, on the one hand, to see this screwed up in random internet discussions where a gold bug argues from well-known observations about the past history of gold. (This is the equivalent of trying to make money at horse-racing by betting on the horse that you think will win - failing to cross the Pons Asinorum.)
But it is surprising is to hear histories of the financial crisis in which prestigious actors argued in crowded auditoriums that, previously, real-estate prices had always gone up, or that previously mortage defaults had been uncorrelated. This is naive inductive reasoning of the sort that only works on falling apples and rising suns and human behavior and everything else in the universe except markets. Shouldn't everyone have frowned and said, "But isn't the marketplace an anti-inductive environment?"
Not that this is standard terminology - but perhaps "efficient market" doesn't convey quite the same warning as "anti-inductive". We would appear to need stronger warnings.
PS: To clarify, the coin example is a humorous exaggeration of what the world would be like if most physical systems behaved the same way as market price movements, illustrating the point, "An exploitable pricing regularity that is easily inducted degrades into inexploitable noise." Here the coin coming up "heads" is analogous to getting an above-market return on a publicly traded asset.