Inverse gambler's fallacy

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The inverse gambler's fallacy is a tempting mistake in judgments of probability, comparable to the gambler's fallacy whence its name derives. Here is an illustration:

You see a pair of fair dice rolled once, and the result is double-sixes. This is a quite improbable result, so you conclude that the dice were probably rolled many times before.

As in the gambler's fallacy, the flaw can be exposed by the slogan "dice have no memory". Each roll of the dice is stochastically independent from each other roll. One roll does not influence another roll. Consequently, even if the dice were rolled many times, that would not make them any more likely to turn up double-sixes for the observed roll. Each roll is on its own.

More generally, the inverse gambler's fallacy states that an improbable event can be made less improbable by the hypothesis that many similar events exist, and that the hypothesis is thence confirmed by the improbable event.

Why are we tempted to reason this way? Probably because of the following consideration. Given many rolls of the dice, it is not unlikely that some roll or another will turn up double sixes. And the more rolls, the more likely it is. But we aren't dealing with the probability that some roll will turn up double sixes; we are dealing with the probability that some specified roll will turn up double sixes. The probability of some improbable result for a specified roll is the same for every other specified roll, and is completely independent of how many rolls there are.

Here is an example that should exorcise any tendency to doubt the preceding:

There is a random number generator with a range from 1 to 100. You are allowed to see one run, and you don't know whether there have been other runs. 17 is the result. Can you reason like so? "17 is a quite improbable result -- one in a hundred. But if there were many runs of the machine, it would be more likely that 17 would result from some run or another. So there were probably many runs of the machine." No, you cannot. The probability of that run turning up 17 is one in a hundred, regardless of how many runs there were. The 'many runs' hypothesis receives no confirmation from the 17-result.

'Special' results increasing the temptation

Here is a parallel example that might tempt you more: Another similar 1–100 random number generator is hooked up so that, if 17 results, it gives the observer fifty dollars. You are allowed to see one run, and you still don't know whether there have been other runs. Lo and behold, 17 is the result. Can you reason like so? "Wow! What do you know! I got 17 – the $50 result! That's one in a hundred odds, how do you like that! I'll bet they've run that machine a lot of times; otherwise, it's too incredible that I happened to get the money". No, you still cannot. The hypothesis of many runs makes it no more likely that 17 would result for that run. It's still one in a hundred.

You might object: "But if there were many runs, somebody or another was bound to get that money, and it might as well have been me. If there was just one run, it's quite unlikely that I would happen to get the money." But this reasoning commits the same error as above. The many runs hypothesis makes it probable that some result or another would be 17. But it doesn't affect the probability that your run would result in 17. It's one in a hundred, regardless of how many runs there were.

We are more tempted to commit the fallacy for an improbable result that is 'special' than for an improbable result that does not stand out from the rest. Why? It is a complicated story.

First, note the difference between 'special' improbable results and 'unremarkable' improbable results. It is not that 'special' results are more improbable. It is that 'special' results can be made more probable by changing our background assumptions. For example, in the 17-cash story, we assumed that the random number generator was truly random. But if we speculate that a benevolent programmer has interfered with the machine, then 17 is more likely than any other number; after all, he wants you to get the money and a 17-result would be the way to do it. In the first and duller 17 story, no such speculation would make 17 more likely than any other number. Since such speculations can make 'special' improbable results more probable, then the speculations are confirmed by 'special' improbable results.

Second, consider the following thoughts about the benevolent-programmer speculation: "Hold on, if there were many runs of the machine, then the programmer would have no reason to choose you over anyone else. The fact that you and not someone else got 17 is still surprising." Assuming that the programmer would maintain the appearance of randomness (and not give everyone 17's), the response is correct. That is, if we know that there have been many runs of the machine, then the benevolent-programmer story does not affect the probability that you get 17. If you walked up to the machine knowing full well that there had been millions of runs, and you happened to get a 17, you would be completely unsurprised at the fact. Note the consequence: the 17-result is no longer 'special', for we can no longer increase its likelihood by appeal to speculative hypotheses that deny our background assumptions.

Finally, remember that, according to the story, we do not know whether there have been many runs. To be sure, if we knew of many runs, the 17-result would not be 'special'; someone was bound to get a 17 and it might as well be you. But given that we don't know of many runs, the 17-result is 'special'; the benevolent-programmer story would make it more probable (as would the story that the program has a bug that biases it to the cash result, and some other stories that deny our background assumptions).

So: why do we commit the fallacy more, where 'special' improbable results are involved? We are confusing a truth with a falsehood that looks the same. The fact is that the hypothesis of many other similar events, if it were known true, would rob the 'special' result of its 'specialness', by making the result insusceptible to assumption-denying hypotheses that make the 'special' event more likely than it would be on our initial assumptions. The falsehood is that 'special' improbable results are made more probable by the hypothesis of many other similar events, and that those results therefore make the hypothesis more likely. Simply put, we confuse "if this hypothesis were true, this phenomenon would be unsurprising" with "this surprising phenomenon makes this hypothesis more likely".

The multiple-world hypothesis

Various lines of reasoning lead to several hypotheses concluding that our universe is not alone, that many "worlds" or all possible worlds exist. See many-worlds interpretation for one sample. The apparent "fine-tuning" that makes our universe suitable for life (see anthropic principle) is sometimes taken to be evidence for many worlds. Hacking (1987) and White (2000) have argued that this latter deduction commits the inverse gambler's fallacy.

The apparent "fine-tuning" is among the arguments for the existence of God, so some proponents of "intelligent design" have misinterpreted that result to mean the multiple worlds hypothesis itself commits the fallacy. Their rationale is that the existence of many worlds vastly increases the probability that some support life, but does not increase the probability that any randomly selected world will support life. However, as the above authors point out, our world is not randomly selected. We obviously can exist only in a world that does support life, so the probability that the world we observe supports life is 1.0.

References

  • Hacking, I. (1987): "The Inverse Gambler's Fallacy: The Argument from Design. The Anthropic Principle Applied to Wheeler Universes." Mind 96: 331-340.
  • White, R. (2000): "Fine-Tuning and Multiple Universes." Nous 34: 260-276.he:כשל_המהמר_ההפוך
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