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I get that a coin toss is independent and multiple of them can't influence each other. However, I always found the Gambler's Fallacy confusing. Empirically the

I get that a coin toss is independent and multiple of them can't influence each other. However, I always found the Gambler's Fallacy confusing. Empirically the coin should should converge to 50% of the flips reading Heads, so why does betting on a Heads not make sense when you've been seeing too many Tails? And yes, I realize that after n flips of Heads, the probability of the n + 1th flip being Heads or Tails is equal. i.e. P(Heads) ^ n * P(Heads) = P(Heads) ^ n * P(Tails), because P(Heads) = P(Tails) = 0.5

arcticbull 5y ago
> ...so why does betting on a Heads not make sense when you've been seeing too many Tails? It doesn't make sense because the events are independent. It's not a better or worse bet to pick one or the other. It doesn't make sense to think one is more than 50% likely because of a series of other independent events when you know the odds a priori.
zarzavat 5y ago
> why does betting on a Heads not make sense when you've been seeing too many Tails? How do you know how many Heads and Tails the coin has flipped in its lifetime? Maybe its previous owner flipped a bunch of Heads. Maybe it flipped some Heads while it was jangling in your pocket. And if you don’t know, how the hell do you expect the coin to know?!
ackbar03 5y ago
Convergence usually means it takes place when something tends to infinity. So yes, if you do it an infinite number of times it does converge to 50%. But the concept of infinity itself is very abstract and is where most people tend to get caught up
pid-1 5y ago
> Empirically the coin should should converge to 50% of the flips reading Heads, The catch is: only for an infinite number of throws and monotonicity is not guaranteed.
bqmjjx0kac 5y ago
> Empirically the coin should should converge to 50% of the flips reading Heads, so why does betting on a Heads not make sense when you've been seeing too many Tails? If you knew, a priori, that 75% of the next 4 flips will land on H, and you've seen THH, you are justified in guessing H. In this scenario, H was actually due. I think it's clear that real coins make no such promises.
foerbert 5y ago
That convergence towards 50% happens at infinity. You aren't flipping the coin an infinite number of times. There is no convergence happening without the infinity. If that doesn't help your intuition, maybe try this technically incorrect view. Those 'extra' Heads you feel are due can appear at any point in the entirety of the future. What are the chances they show up in the next five minutes, never mind the next billion years? You're looking for a handful events spread over the entire rest of time itself, after all.
- kllvql 5y ago
  I found myself struggling with the intuition of this similarly to amingilani until I made the connection you're mentioning to infinity. The thing that helped me understand the fallacy was understanding how truly large infinity is. If you've seen 10 coin flips, or even 10,000 or 10,000,000, those are all effectively nothing when compared against infinity. If you know ahead of time that a coin is fair (so you're not trying to test this hypothesis), you gain zero information about the long-term behavior of the coin from any finite number of flips. As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.
- horsawlarway 5y ago
  > There is no convergence happening without the infinity. This is wrong. It takes considerably fewer flips than "infinity" to see distinct convergence here. If the coin is truly even - you have a binomial distribution, and the probability that you are far out to either edge of that distribution (nearly all heads, or nearly all tails) is vanishingly small with just 1000 flips.
  - foerbert 5y ago
    Probably, yes. But it's not a guarantee until you include infinity. And that's the important distinction here. Without any such guarantee of the larger result, all you have is a 50/50 chance for the next flip. With infinity, you can pick out a portion of the sequence with an unusual number of Tails and expect there to be an offsetting number of Heads elsewhere in the full (infinite) sequence.
s1artibartfast 5y ago
>the coin should should converge to 50% of the flips reading Heads This is incorrect. The convergence is to: (current heads + 0.5(n flips)/ (current flips + n flips). For a gambler flipping coins, If they are neutral at the time of prediction, you expect them to end neutral. If they are up flips when you enter the room, you expect them to the run up flips.
hamstergene 5y ago
From the practical standpoint, it actually makes more sense to bet on tails, because it may be a symptom of unfair coin :) If the coin is fair the choice is unimportant, if unfair you win.
panarky 5y ago
It helps me to write a simple program to simulate this. A random number <=0.5 is heads, >0.5 is tails. Flip 100 million coins. After you get 10 of the same result in a row, count how many times the 11th is the same as the first 10.
r3trohack3r 4y ago
Given a fair coin: The odds of getting 12 Heads in a row is quite low. The odds of getting 12 Heads in a row when you’ve already observed 11 coin flips land heads up is 50%. Knowing that the odds of 12 Heads in a row is low, that you’ve seen 11 Heads in a row, and that - if you observe the next flip land heads up - you’ll have observed something quite rare, doesn’t influence the next coin flip.
samatman 4y ago
For the same reason that a Bitcoin block is mined every ten minutes: at any given moment, it's going to take about ten minutes to find the right hash, and it doesn't matter if the last one was a nanosecond ago or 12 minutes ago.
aidenn0 4y ago
Perhaps this will help: The odds of getting the sequence HTTHT are exactly the same as the odds of getting HHHHH, and similarly the odds of getting HTTHTH are exactly the same as HHHHHH. Another way of thinking of it is that there's roughly a one in a thousand chance of getting 10 heads in a row; there's a one in two-thousand chance of getting 11 heads in a row, but if you've already gotten 10 heads in a row, you've already ruled out the ~1998/2000[1] possibilities that can't ever result in 11 heads (because they didn't start out with 10 heads in a row), so you only need to rule out one of the ~2/2000 remaining possibilities. 1: Actual numbers is 2046/2048 and 2/2048
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arcticbull 5y ago

> ...so why does betting on a Heads not make sense when you've been seeing too many Tails? It doesn't make sense because the events are independent. It's not a better or worse bet to pick one or the other. It doesn't make sense to think one is more than 50% likely because of a series of other independent events when you know the odds a priori.

zarzavat 5y ago

> why does betting on a Heads not make sense when you've been seeing too many Tails? How do you know how many Heads and Tails the coin has flipped in its lifetime? Maybe its previous owner flipped a bunch of Heads. Maybe it flipped some Heads while it was jangling in your pocket. And if you don’t know, how the hell do you expect the coin to know?!

ackbar03 5y ago

Convergence usually means it takes place when something tends to infinity. So yes, if you do it an infinite number of times it does converge to 50%. But the concept of infinity itself is very abstract and is where most people tend to get caught up

pid-1 5y ago

> Empirically the coin should should converge to 50% of the flips reading Heads, The catch is: only for an infinite number of throws and monotonicity is not guaranteed.

bqmjjx0kac 5y ago

> Empirically the coin should should converge to 50% of the flips reading Heads, so why does betting on a Heads not make sense when you've been seeing too many Tails? If you knew, a priori, that 75% of the next 4 flips will land on H, and you've seen THH, you are justified in guessing H. In this scenario, H was actually due. I think it's clear that real coins make no such promises.

foerbert 5y ago

That convergence towards 50% happens at infinity. You aren't flipping the coin an infinite number of times. There is no convergence happening without the infinity. If that doesn't help your intuition, maybe try this technically incorrect view. Those 'extra' Heads you feel are due can appear at any point in the entirety of the future. What are the chances they show up in the next five minutes, never mind the next billion years? You're looking for a handful events spread over the entire rest of time itself, after all.

kllvql 5y ago
I found myself struggling with the intuition of this similarly to amingilani until I made the connection you're mentioning to infinity. The thing that helped me understand the fallacy was understanding how truly large infinity is. If you've seen 10 coin flips, or even 10,000 or 10,000,000, those are all effectively nothing when compared against infinity. If you know ahead of time that a coin is fair (so you're not trying to test this hypothesis), you gain zero information about the long-term behavior of the coin from any finite number of flips. As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.
horsawlarway 5y ago
> There is no convergence happening without the infinity. This is wrong. It takes considerably fewer flips than "infinity" to see distinct convergence here. If the coin is truly even - you have a binomial distribution, and the probability that you are far out to either edge of that distribution (nearly all heads, or nearly all tails) is vanishingly small with just 1000 flips.
- foerbert 5y ago
  Probably, yes. But it's not a guarantee until you include infinity. And that's the important distinction here. Without any such guarantee of the larger result, all you have is a 50/50 chance for the next flip. With infinity, you can pick out a portion of the sequence with an unusual number of Tails and expect there to be an offsetting number of Heads elsewhere in the full (infinite) sequence.

s1artibartfast 5y ago

>the coin should should converge to 50% of the flips reading Heads This is incorrect. The convergence is to: (current heads + 0.5(n flips)/ (current flips + n flips). For a gambler flipping coins, If they are neutral at the time of prediction, you expect them to end neutral. If they are up flips when you enter the room, you expect them to the run up flips.

hamstergene 5y ago

From the practical standpoint, it actually makes more sense to bet on tails, because it may be a symptom of unfair coin :) If the coin is fair the choice is unimportant, if unfair you win.

panarky 5y ago

It helps me to write a simple program to simulate this. A random number <=0.5 is heads, >0.5 is tails. Flip 100 million coins. After you get 10 of the same result in a row, count how many times the 11th is the same as the first 10.

r3trohack3r 4y ago

Given a fair coin: The odds of getting 12 Heads in a row is quite low. The odds of getting 12 Heads in a row when you’ve already observed 11 coin flips land heads up is 50%. Knowing that the odds of 12 Heads in a row is low, that you’ve seen 11 Heads in a row, and that - if you observe the next flip land heads up - you’ll have observed something quite rare, doesn’t influence the next coin flip.

samatman 4y ago

For the same reason that a Bitcoin block is mined every ten minutes: at any given moment, it's going to take about ten minutes to find the right hash, and it doesn't matter if the last one was a nanosecond ago or 12 minutes ago.

aidenn0 4y ago

Perhaps this will help: The odds of getting the sequence HTTHT are exactly the same as the odds of getting HHHHH, and similarly the odds of getting HTTHTH are exactly the same as HHHHHH. Another way of thinking of it is that there's roughly a one in a thousand chance of getting 10 heads in a row; there's a one in two-thousand chance of getting 11 heads in a row, but if you've already gotten 10 heads in a row, you've already ruled out the ~1998/2000[1] possibilities that can't ever result in 11 heads (because they didn't start out with 10 heads in a row), so you only need to rule out one of the ~2/2000 remaining possibilities. 1: Actual numbers is 2046/2048 and 2/2048

deleted 4y ago

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