The Gambler's Fallacy is Not a Fallacy

(2900 words; 15 minute read.)

[5/11 Update: Since the initial post, I've gotten a ton of extremely helpful feedback (thanks everyone!). In light of some of those discussions I've gone back and added a little bit of material. You can find it by skimming for the purple text.]
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[5/28 Update: If I rewrote this now, I'd now reframe the thesis as: "Either the gambler's fallacy is rational, or it's much less common than it's often taken to be––and in particular,  standard examples used to illustrate it don't do so."]

​A title like that calls for some hedges––here are two.  First, this is work in progress: the conclusions are tentative (and feedback is welcome!). Second, all I'll show is that rational people would often exhibit this "fallacy"––it's a further question whether real people who actually commit it are being rational.

Off to it.

On my computer, I have a bit of code call a "koin". Like a coin, whenever a koin is "flipped" it comes up either heads or tails. I'm not going to tell you anything about how it works, but the one thing everyone should know about koins is the same thing that everyone knows about coins: they tend to land heads around half the time.

I just tossed the koin a few times. Here's the sequence it's landed in so far:
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T H T T T T T

How likely do you think it is to land heads on the next toss?  You might look at that sequence and be tempted to think a heads is "due", i.e. that it's more than 50% likely to land heads on the next toss. After all, koins usually land heads around half the time––so there seems to be an overly long streak of tails occurring.

But wait! If you think that, you're committing the gambler's fallacy: the tendency to think that if an event has recently happened more frequently than normal, it's less likely to happen in the future. That's irrational.  Right?

Wrong.  Given your evidence about koins, you should​ be more than 50% confident that the next toss will land heads; thinking otherwise would be a mistake.

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