Gambler's Fallacy Revisited

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alan1000
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Gambler's Fallacy Revisited

Post by alan1000 »

It's often said that "the dice have no memory". In ten coin tosses, if the first 9 come up heads, it doesn't make it any more likely that the tenth throw will come up tails; the probability remains 50% or, as the statisticians prefer to say, 0.5.

This is provably true for a limited sample of 10 tosses. But does it remain true for an infinite number of tosses?

Opinion polls usually try to get at least 1000 responses because they know that the more responses you get, the more reliable the conclusions you can draw from it. Given ever-larger samples, the true average tends to assert itself.

So, let us suppose the gambler intends to remain in the game for a (theoretically) infinite number of throws. In a million tosses of the coin, all things being equal, we would expect that half would come up heads and half tails. So if the last 10 have come up heads, we would expect that at some future stage this would be balanced with a series of 10 tails. And the future begins with the next toss.

If the gambler a) intends to stay in the game for an infinite period, and b) is reasoning inductively, then, it would be wise to assume that if the last 9 throws have come up heads, the probability is increased that the next throw will be tails. Is this a reasonable assumption?
Iwannaplato
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Re: Gambler's Fallacy Revisited

Post by Iwannaplato »

No. There is no causal connection between the earlier throws and later ones. The universe is not noticing the difference between a coin tossed in the air and the same coin sitting in your pocket. And string of throws is just some stuff that happens in the middle of the existence of a coin. We can't participate in an infinite number of throws and half of infinity is infinity, so we really can't use infinite throws to predict anything or apply it to a real life situation. There's a statistical tendency which is different from some kind of causal tendency.
bobmax
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Re: Gambler's Fallacy Revisited

Post by bobmax »

It is necessary that the heads and tails are equal to infinity.

And it is precisely this necessity that requires that at each toss of the coin the probability is 0.5.

If 9 heads in a row, I still have infinity in front of me, which keeps the probability of the next coin toss fixed at 0.5.

The infinite obliges even though it does not exist, the finite exists but does not oblige in anything.
Iwannaplato
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Re: Gambler's Fallacy Revisited

Post by Iwannaplato »

bobmax wrote: Wed Oct 05, 2022 4:29 pm It is necessary that the heads and tails are equal to infinity.

And it is precisely this necessity that requires that at each toss of the coin the probability is 0.5.

If 9 heads in a row, I still have infinity in front of me, which keeps the probability of the next coin toss fixed at 0.5.

The infinite obliges even though it does not exist, the finite exists but does not oblige in anything.
Yes, or another way to put it - and I liked yours - is half of infinity is infinity.

So on any given toss, we have an infinite number of heads and an infinite number of tails coming. Who knows what the next toss will be. But we know the odds on the next toss and the next. And that never changes.
Impenitent
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Re: Gambler's Fallacy Revisited

Post by Impenitent »

flipping coins?

nine heads in a row are on Hydra - run away...

-Imp
Age
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Re: Gambler's Fallacy Revisited

Post by Age »

alan1000 wrote: Wed Oct 05, 2022 2:58 pm It's often said that "the dice have no memory". In ten coin tosses, if the first 9 come up heads, it doesn't make it any more likely that the tenth throw will come up tails; the probability remains 50% or, as the statisticians prefer to say, 0.5.

This is provably true for a limited sample of 10 tosses. But does it remain true for an infinite number of tosses?
Why would it NOT?
alan1000 wrote: Wed Oct 05, 2022 2:58 pm Opinion polls usually try to get at least 1000 responses because they know that the more responses you get, the more reliable the conclusions you can draw from it. Given ever-larger samples, the true average tends to assert itself.
But what the opinions are of 'you', human beings, NOR if the opinions are the same or similar, this has absolutely NOTHING AT ALL to do with the toss of a coin.
alan1000 wrote: Wed Oct 05, 2022 2:58 pm So, let us suppose the gambler intends to remain in the game for a (theoretically) infinite number of throws. In a million tosses of the coin, all things being equal, we would expect that half would come up heads and half tails.
'you' may well EXPECT that, but 'I' CERTAINLY DO NOT.
alan1000 wrote: Wed Oct 05, 2022 2:58 pm So if the last 10 have come up heads, we would expect that at some future stage this would be balanced with a series of 10 tails.
If that is what 'you' EXPECT, then so be it. But 'I' would NOT EXPECT 'this'.

Also, have what 'you' EXPECTED previously EVER NOT come to fruition?
alan1000 wrote: Wed Oct 05, 2022 2:58 pm And the future begins with the next toss.
The 'future', just like the 'end', BEGINS right HERE, right NOW.

The 'future' does NOT BEGIN with the NEXT toss of a coin, NOR ANY thing else.
alan1000 wrote: Wed Oct 05, 2022 2:58 pm If the gambler a) intends to stay in the game for an infinite period, and b) is reasoning inductively, then, it would be wise to assume that if the last 9 throws have come up heads, the probability is increased that the next throw will be tails.
The REASON WHY so-called "gamblers" are NOT 'rich', with money, is BECAUSE they place their bets on ASSUMPTIONS.

AND, ALL ASSUMPTIONS are NOT necessarily true, right, NOR correct AT ALL. This is WHY those, essentially, just GUESSES are called ASSUMPTIONS.

Also, one could ASSUME absolutely ANY thing here, BUT the probability that the next toss of a coin is heads or tails will ALWAYS BE 50%

alan1000 wrote: Wed Oct 05, 2022 2:58 pm Is this a reasonable assumption?
'you', adult human beings, can "reason-out" just absolutely ANY 'thing'. And, this ability helps in the EXPLAINING OF WHY 'you' are SO Wrong, SO OFTEN.
Age
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Re: Gambler's Fallacy Revisited

Post by Age »

bobmax wrote: Wed Oct 05, 2022 4:29 pm It is necessary that the heads and tails are equal to infinity.

And it is precisely this necessity that requires that at each toss of the coin the probability is 0.5.

If 9 heads in a row, I still have infinity in front of me, which keeps the probability of the next coin toss fixed at 0.5.

The infinite obliges even though it does not exist, the finite exists but does not oblige in anything.
What PROOF do you have that the so-called 'infinite' does NOT exist?

Or is this just an ASSUMPTION of yours?
wtf
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Re: Gambler's Fallacy Revisited

Post by wtf »

alan1000 wrote: Wed Oct 05, 2022 2:58 pm If the gambler a) intends to stay in the game for an infinite period, and b) is reasoning inductively, then, it would be wise to assume that if the last 9 throws have come up heads, the probability is increased that the next throw will be tails. Is this a reasonable assumption?
What would you say is the mechanism by which a coin I take out of my pocket, remembers its history of flips?

And how does the coin know whether you intend to flip it finitely or infinitely many more times in the future, assuming for sake of discussion that you could do the latter?
Skepdick
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Re: Gambler's Fallacy Revisited

Post by Skepdick »

alan1000 wrote: Wed Oct 05, 2022 2:58 pm It's often said that "the dice have no memory". In ten coin tosses, if the first 9 come up heads, it doesn't make it any more likely that the tenth throw will come up tails; the probability remains 50% or, as the statisticians prefer to say, 0.5.

This is provably true for a limited sample of 10 tosses. But does it remain true for an infinite number of tosses?

Opinion polls usually try to get at least 1000 responses because they know that the more responses you get, the more reliable the conclusions you can draw from it. Given ever-larger samples, the true average tends to assert itself.

So, let us suppose the gambler intends to remain in the game for a (theoretically) infinite number of throws. In a million tosses of the coin, all things being equal, we would expect that half would come up heads and half tails. So if the last 10 have come up heads, we would expect that at some future stage this would be balanced with a series of 10 tails. And the future begins with the next toss.

If the gambler a) intends to stay in the game for an infinite period, and b) is reasoning inductively, then, it would be wise to assume that if the last 9 throws have come up heads, the probability is increased that the next throw will be tails. Is this a reasonable assumption?
The trouble with your framing is that you have no idea what you mean by the words "provably" and "true", and so you have absolutely no idea what you are asking.

On the one hand you are reasoning empirically - do the experiment. Make measurements. Do calculations.
On the other hand you are reasoning theoretically - using infinitary processes and the language of probability theory which produces only tautologies.

Are you asking an empirical question or a theoretical question?

Every possible sequence of coin toss outcomes appears in an infinite game.
10 heads followed by 10 tails is just as likely a sequence as a googol heads followed by a googol tails.

In fact n of tails followed by n of heads is just as likely as n of tails followed by another n of tails. For any n.

At some point your vocabulary is going to have to start talking about periodicities and frequencies.
Last edited by Skepdick on Thu Oct 06, 2022 9:18 am, edited 2 times in total.
Skepdick
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Re: Gambler's Fallacy Revisited

Post by Skepdick »

wtf wrote: Thu Oct 06, 2022 7:59 am What would you say is the mechanism by which a coin I take out of my pocket, remembers its history of flips?
In a deterministic universe that is a stupid question. The exact state of the coin at any given point in time is a direct product of its entire history.

That you are unable to recall/reconstruct this history is your epistemic problem. The coin's ontology doesn't care.
bobmax
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Re: Gambler's Fallacy Revisited

Post by bobmax »

Iwannaplato wrote: Wed Oct 05, 2022 6:47 pm
bobmax wrote: Wed Oct 05, 2022 4:29 pm It is necessary that the heads and tails are equal to infinity.

And it is precisely this necessity that requires that at each toss of the coin the probability is 0.5.

If 9 heads in a row, I still have infinity in front of me, which keeps the probability of the next coin toss fixed at 0.5.

The infinite obliges even though it does not exist, the finite exists but does not oblige in anything.
Yes, or another way to put it - and I liked yours - is half of infinity is infinity.

So on any given toss, we have an infinite number of heads and an infinite number of tails coming. Who knows what the next toss will be. But we know the odds on the next toss and the next. And that never changes.
Yes infinity is tremendous.
The likelihood of a million heads in a row is very rare indeed.
But once this very rare sequence has taken place, we will still find ourselves in front of all the infinite combinations that contain a million heads in a row.
wtf
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Re: Gambler's Fallacy Revisited

Post by wtf »

bobmax wrote: Thu Oct 06, 2022 7:27 pm Yes infinity is tremendous.
The likelihood of a million heads in a row is very rare indeed.
But once this very rare sequence has taken place, we will still find ourselves in front of all the infinite combinations that contain a million heads in a row.
Indeed, in an infinite sequence of flips, the probability is zero that you will not get a million heads in a row at some point. The probability is 1 that you will.
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Agent Smith
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Re: Gambler's Fallacy Revisited

Post by Agent Smith »

There's a paradox associated with the so-called Gambler's fallacy: :shock: vs. :|
alan1000
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Re: Gambler's Fallacy Revisited

Post by alan1000 »

Iwannaplato wrote: Wed Oct 05, 2022 3:48 pm No. There is no causal connection between the earlier throws and later ones. The universe is not noticing the difference between a coin tossed in the air and the same coin sitting in your pocket. And string of throws is just some stuff that happens in the middle of the existence of a coin. We can't participate in an infinite number of throws and half of infinity is infinity, so we really can't use infinite throws to predict anything or apply it to a real life situation. There's a statistical tendency which is different from some kind of causal tendency.
So how do you explain the fact that, given a sufficiently large sample, the mean tends to approach closer and closer to a true average?
Skepdick
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Re: Gambler's Fallacy Revisited

Post by Skepdick »

alan1000 wrote: Wed Nov 30, 2022 4:19 pm So how do you explain the fact that, given a sufficiently large sample, the mean tends to approach closer and closer to a true average?
Why does it require an explanation? It's a tautology of an infinitary definition.

The key word being "infinitary". If you can repeat the experiment over and over, with the same conditions an infinite number of times...

But you can't.
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