a bonobo humanity?

Humans are capable of reasoning, but not always or often very well. Daniel Kahneman’s famous book Thinking, fast and slow provides us with many examples, and not being much of a clear thinker myself, where probability and all that Bayesian stuff is concerned, I’ll start with something really simple before ascending, one day, to the simply simple. And not being much of a gambler, I’d never heard of the gambler’s fallacy before. It appears to be a simple and obvious fallacy, but I’m sure I can succeed in making it more confusing than it should be.

The fallacy involves believing that what has occurred before might dictate what happens in the future, in a particular context. It’s best explained by the tossing of a coin. With a fair coin, the probability of it landing tails up, on any toss, is .5, given that, in probability language, absolute certainty is given a value of 1, and no possibility at all is given 0. The key here is what I’ve italicised – the fallacy lies in believing that the coin, as if it’s a thinking being, has an interest in maintaining a result, over many tosses, of 50% tails – so that if results skew towards zero, say after 6 heads results in a row, the probability of the next toss being tails will rise above .5. 

Put another way: assuming a fair coin, the probability of it landing heads on one toss is .5. That should mean that over time, with x number of tosses, assuming x to be a very large number, the result for a heads should approach 50%. So it would seem quite reasonable, if you were keeping count, to bet on a result that brings the average closer to 50%. That’s without imagining that the coin wants to get to 50%. It just should, shouldn’t it?

The clear answer is no. There can be no influence from the past on any new coin toss. How can there be? That would be truly weird if you think about it. The overall results may approach 50%, according to the law of large numbers, but that’s independent of particular tosses. If you look at it this way, creating a dependency, you decide to bet on a pair of tosses. It could be HH, TT, HT or TH. Those are the only four options and the probability of each of them is .25 (i.e .5 x .5). So you might think that, after two heads in a row, it would be wise to bet on tails. But this bet would still have a .5 probability of succeeding, and the result HHT, taken together, would be .5 x .5 x .5, which is .125 or one eighth, the same as all the other seven results of three coin tosses. The probability doesn’t change before each toss, no matter the result of the previous toss. 

So far, so clear, but it would be hard not to be influenced into betting against a run continuing. That’s not irrational, is it? But nor is it rational, considering there’s alway a 50/50 chance with each toss. It’s just a bet. And yet… I’m reminded of Swann in a A la recherche du temps perdu, as my mind clouds over…