Let's play a game. We're going to flip a coin, you get to call heads or tails, and you get $1 if you win and pay me $1 if you lose.
Pretty straightforward, yes?
We flip the coin a bunch. It comes down roughly half heads and half tails.
Then, suddenly the coin comes down heads six times in a row.
I say, "Hold on, the coin is due for tails. I'm not taking a $1 vs. $1 bet here... I need you to put up $1.20 if you want to bet on tails."
Do you take that bet?
Okay. The coin comes down twenty times on tails in a row. Now it's really due for heads, right?
So, would you pay more than even odds to bet on heads?
(I hope not)
The gambler's fallacy is the idea that past, independent outcomes affect future outcomes. Because a coin will come down heads half the time and tails half the time, if you have a huge run of heads in a row... then tails must be "due", right?
This fallacy has lost a lot of people a lot of money.
If you're playing poker, your chance of getting the cards you want dealt to you are exactly the normal chances in any given hand. It doesn't matter how the game has been going the last 10 hands, 20 hands, or even the last 10 hours.
Same with flipping coins.
Same with anything with independent outcomes.
Now, sometimes outcomes aren't independent. If you know a "streaky" person who goes through manic phases and long slumps, you might predict a slump if the streaker has been on an extraordinarily long manic phase.
The key is figuring out whether events are independent or not. If they are - like coins or flipping cards - then nothing is ever "due," and wagering like something is "due" is a really bad idea.
Cards and flipping coins are obvious to see this on, but there's lots of real life applications. If you've been having a run of probability coming down favorably to you, it doesn't mean "bad luck" is due. If you've been having a run of bad probability, it doesn't change the chances of a good outcome.
If there's really a 50% chance of something happening, then it doesn't matter if it just happened 10 times in a row the other way. Random is actually rather quite random. You should probably think if your probability estimates are off if you see a ton of data in the other direction. And maybe the recent past isn't independent from the future - an athlete with bad performance lately might be injured or have screwed up mechanics.
But again, always - if you've got independent events, nothing is due, ever. Don't bet on something being "due" with independent outcomes or you're going to risk getting burnt.
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