Read about the gambler’s fallacy here. The basic idea is this: it is false to think that, if a fair coin is flipped 10 times in row and tails comes up each time, heads will be more likely on the 11th flip.
If the coin is truly fair, the chances will be 50/50 every single time. So 11th flip will be 50/50 too.
It strikes me that those who call this a fallacy misunderstand those who think that the 11th is more likely to be heads than tails.
If you flip a coin a hundred times and every time it comes up tails, isn’t this good evidence that the coin isn’t a fair coin?
Sure, it is logically possible that tails will be flipped 100 times in row. But it has never happened to me or (probably) anyone you know. And if it did happen you’d be very suspicious that something funny was going on. Admit it. You’d be suspicious.
So while the 100th flip is still 50/50, that’s besides the point. What makes the head seem more likely after 99 tails is the joint meaning of the 100th flip together with the previous 99 flips. 100 tails in a row with a fair coin is unlikely, i.e. we don’t expect it to happen and if it did happen we’d be suspicious. (This is true even as we acknowledge that each flip has 50/50 chance of being tails.)
If the coin really is fair, then sooner or later heads will start being flipped. And, given that the coin is fair, we expect it will happen sooner rather than later. So heads seems more likely than tails.
The standard response to this is to admit that, before any coins have been flipped, the chances of 100 tails is low. But once some flips have been made the probability needs to be re-evaluated in light of these flips. So after 99 tails have been flipped, the chances of 100 tails in a row is considerably higher than before. But the fact that 100 tails is now more likely than before doesn’t mean the 100th is more likely than before to be tails. The chances are still 50/50.
But this completely misses the point. No one will deny that every flip of a fair coin is 50/50. Heads seems more likely because of its joint meaning together with other 99 tails flipped. This has nothing to do with the 100th flip itself. It has everything to do with its role in providing what is necessary for a very unlikely event - 100 straight tails. If 100 straight tails is indeed unlikely (and everyone admits this), then the conditions which would obtain this must be unlikely, right? For if the conditions were likely, then the 100 straight tails would be likely. But it is not. (Remember, we aren’t talking about these conditions taken in isolation, but according to their joint meaning as parts of a whole.)
As I see things, it all comes back to this: those who think that the gambler’s fallacy really is a fallacy think that rationality demands we consider each flip in isolation from the joint meaning it provides to a greater whole. But why think that? The standard response (see above) doesn’t give us a reason to think that. So don’t bother repeating the standard response to me. I don’t doubt that each flip is 50/50. I just think this has nothing to do with our rational and justifiable shock in the face of an ever increasing run of tails in a row.
Evidently I believe that it is perfectly consistent to believe: (a) that a flip is 50/50 and (b) it is also more likely to be a either a head or tail given its joint meaning in a greater whole.
Show me why I’m wrong.