Why does something like the gambler’s fallacy hold true in an instance like the monte carlo roulette incident?

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In case you’re wondering what the monte carlo incident is, it was a game of roulette that landed on black 26 times in a row… the odds of that happening is 1 in 66.6 million

The gamblers fallacy is a fallacy that people who gamble tend to think if something has a long streak it’s going to change.

If the odds of it being 26 blacks in a row is 1 in 66.6 million why would that be a fallacy? Obviously it could always be a 27th black but thats incredibly unlikely and statistically speaking red would be far more likely…

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Red is no more likely than black for the 27th spin. There’s slightly less than a 50% chance that it lands on Red and a slightly less than 50% chance it lands on black(thanks to the Green zeros)

The odds of getting to that pattern are 1 in 66 million, but the odds on the next spin are always the same

If you do a few billion roulette spins, you’ll likely have at least one string of 30 blacks in a row and another string of 30 reds in a row, simply because you have a large enough sample that suddenly unlikely combinations become near certainty. But just because I know that somewhere in the billions of spins will be 30 blacks in a row, that doesn’t mean I should go all in on the first one to reach 29 in a row.

The gamblers fallacy would still hold true, they thought the odds of a red would be likely.
On paper the odds of the next spin are unchanged, despite what the gamblers think.
The unlikely hood is dispensed with in history and history doesn’t matter to the odds of the next spin, at least in theory.

But in practice however, think of it another way:
You picked up a pencil and let it go.
1st time, it falls to the ground.
2nd time, it falls to the ground.
3rd time…
28th time, it falls to the ground.

Predict what would happen for the 29th time you picked up a pencil and let it go.

Terrible pattern recognition on that time would be “uhhh, it will rocket up to the ceiling, because it would break the streak”, even ” there are even odds it will rocket to the ceiling or fall to the floor” is lousy critical thinking.

Apparently you have to consider the 26 straight reds against an infinite number of spins. It looked weird but wasn’t necessarily a miracle.

The gambler’s fallacy isn’t that a long streak is unlikely. It’s that the probability of the next spin being black changes based on previous spins, that the probability of red on the next spin is 60% or 70% instead of just 50%.

In reality, the outcome of any one spin is independent of all prior spins. If I give you a wheel that has just gotten 25 blacks and one that has just gotten a mix of 25 reds and blacks, each still has the same chance of getting black on the next spin.

It’s just unlikely for the 25 black streak to have already happened on the all-black wheel.

The chance of *any* unique pattern of 25 prior spins — say, red-black-red-black-red-black — is equally fantastically improbable. We just think of all black as being interesting because it happens to stand out to us.

Imagine you walk up to a roulette table. You haven’t seen any of the previous spins and don’t know what was previously hit. You decide to place a bet on black thinking that your odds are slightly less than 50% (because, numerically, they are). They spin the wheel and you’re waiting to see what the ball lands on when a guy walks up and says, “it’s so weird, it landed on black 26 times before this.” Because he told you that, are you less likely to win this particular spin? No, your odds did not change because this guy told you what happened in the past. Each spin is completely independent of all previous and future spins.