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Old 16th September 2009, 11:28 PM   #41
shuttlt
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The OP specifically rules out spinning coins by saying that the coin is caught. The (flawed) argument in the OP is not based on the coin being biased.
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Old 17th September 2009, 03:15 AM   #42
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Originally Posted by shuttlt View Post
I regretted the whole position it right thing after writing it and should have corrected my example. I didn't mean a dishonest flip, I just got side tracked into thinking about the mechanics of how a biased coin might work. Please ignore my nonsense about positioning the coin.
No worries

Originally Posted by shuttlt View Post
What are your reasons for believeing that there is, in practice, no such thing as a biased coin. I had always assumed they existed.
I think you've misunderstood me. I'm saying that every coin is biased to some degree. But I'm also saying that it makes absolutely no difference to the likely outcome of a fair toss.

Originally Posted by shuttlt View Post
In any case, I don't quite know how I came to be arguing with you that a biased coin is possible since our original disagreement had nothing to do with that. I was claiming that the argument in the OP was flawed because the odds of a coin flipping 6 times are not the same as the odds of a coin flipping 7 times. You asked on what basis I thought this and somehow now we are talking about whether or not biased coins exist.
Yes, I think we've digressed unnecessarily, and don't really hold any significant difference in views.
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Old 17th September 2009, 07:22 AM   #43
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Back to the actual paper, I think it's quite an interesting result and a very simple analysis - I wish I'd thought of it!

It would be fun to figure out how to flip coins to increase the bias. It doesn't sound very hard to do and should be almost impossible to detect by eye.
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Old 17th September 2009, 08:18 AM   #44
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The argument in the paper seems a lot more complicated than I had realized from the OP.
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Old 17th September 2009, 08:36 AM   #45
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Originally Posted by shuttlt View Post
The argument in the paper seems a lot more complicated than I had realized from the OP.
Here's a summary:

Originally Posted by sol invictus View Post
I think the paper is correct based on a quick reading of the first few pages (and the fact that at least one of the authors is an extremely good mathematician and a former magician). But the argument made in the paper bears zero resemblance to the argument in the blog quoted in the OP.

Here's the argument made in the paper. When you flip a coin you impart some angular momentum to it. Consider two cases:

(1) you flip the coin so it spins around a horizontal axis

(2) you flip it so it spins like a tossed pizza

In case (1) the angular momentum vector (that's the axis it spins around) is perpendicular to the vertical, in case (2) it's parallel.

What they find is that case (1) is unbiased in the limit the coin has time to flip many times, and in case (2) the result is completely biased (because the coin doesn't flip at all).

What about cases in between? Not surprisingly, they're in between those two cases, in other words the coin is biased a bit. After observing some actual coin flips to get some data, they conclude the flip is biased at about 51%.

Sounds right to me.
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Old 17th September 2009, 10:20 AM   #46
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Wow, totally different to the OP.
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Old 17th September 2009, 12:02 PM   #47
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The paper is very different to the OP (which most of us were responding to).

They put together quite a nice detailed physical model of the coin toss - which seems impressive - but then somewhat ruin it by making pretty crude assumptions about the probability distributions of the inputs to this model.

I say this because the output distributions will merely be a function of the input distributions, and the subtle effects they are looking for could be swayed by these assumptions.

I was also surprised that 1 in 25 of their "vigorous" flips did not flip the coin once. This brings me back to an earlier point about the statistics being a function of the flipper. I don't think this would be anything like (for example) if I were the one flipping the coin!

It is an interesting paper, as far as the assumptions go. I think I would need rather more convincing evidence of those assumptions before buying into the predictions made by their model.
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Old 17th September 2009, 12:10 PM   #48
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I'd be interested in seeing the results of a large number of real world coin tosses. Do they show any bias?
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Old 17th September 2009, 01:58 PM   #49
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Originally Posted by shuttlt View Post
I'd be interested in seeing the results of a large number of real world coin tosses. Do they show any bias?
The paper itself quotes a study in which they praise someone for their perseverance in completing 10,000 coin tosses. Unfortunately, the sampling error is too great, and not accurate enough to detect the effect they describe. (The study found no statistically significant bias in the results)

The paper estimated that a total of 250,000 tosses would be needed to confidently identify a bias of the magnitude they describe.

They appear unwilling to embark on such a study! They need to make an event of it - go for a world record

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Old 17th September 2009, 02:06 PM   #50
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Originally Posted by CoolSceptic View Post
They put together quite a nice detailed physical model of the coin toss - which seems impressive - but then somewhat ruin it by making pretty crude assumptions about the probability distributions of the inputs to this model.
They're mathematicians.

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I say this because the output distributions will merely be a function of the input distributions, and the subtle effects they are looking for could be swayed by these assumptions.
Yes, but I think the only flips that are unbiased are ones in which all the angular momentum goes into flipping around a horizontal axis, and none goes into spinning the coin like a pizza. Surely any real flip will have a little of both, and if so I think that means any real flip will be biased at least a little.
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Old 17th September 2009, 02:15 PM   #51
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Originally Posted by sol invictus View Post
They're mathematicians.
Have I missed something here? I'm not sure what you're getting at.

Originally Posted by sol invictus View Post
Yes, but I think the only flips that are unbiased are ones in which all the angular momentum goes into flipping around a horizontal axis, and none goes into spinning the coin like a pizza. Surely any real flip will have a little of both, and if so I think that means any real flip will be biased at least a little.
No, that certainly does not follow. Just because we know the values at point A and B - say f(A) and f(B) - does not automatically imply that f(C) will lie between f(A) and f(B) just because C lies between A and B. And, indeed, the paper makes no such broad-brush claim either.
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Old 17th September 2009, 06:55 PM   #52
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Originally Posted by shuttlt View Post
Wow, totally different to the OP.
Originally Posted by CoolSceptic View Post
The paper is very different to the OP (which most of us were responding to).
When you say "OP", I assume that you're not really referring to my original post, you're referring to the article by James Devlin, which I quoted in the OP. Remember, I warned you that "James Devlin says this, parts of which may or may not be supported by the paper."
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Old 17th September 2009, 08:10 PM   #53
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Originally Posted by CoolSceptic View Post
Have I missed something here? I'm not sure what you're getting at.
Just that mathematicians aren't necessarily the best people to be doing experiments.

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No, that certainly does not follow.
Yes, it certainly follows.

Quote:
Just because we know the values at point A and B - say f(A) and f(B) - does not automatically imply that f(C) will lie between f(A) and f(B) just because C lies between A and B. And, indeed, the paper makes no such broad-brush claim either.
What are you talking about? Did you even glance at the paper before making stupid pronouncements? That's the main result of the paper, what the entire effect rests on. It's "Theorem 2", top of page 3.

Last edited by sol invictus; 17th September 2009 at 08:12 PM.
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Old 18th September 2009, 01:34 AM   #54
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Originally Posted by sol invictus View Post
Just that mathematicians aren't necessarily the best people to be doing experiments.
OK, that's what I thought you meant.

Originally Posted by sol invictus View Post
Yes, it certainly follows.

What are you talking about? Did you even glance at the paper before making stupid pronouncements? That's the main result of the paper, what the entire effect rests on. It's "Theorem 2", top of page 3.
Just to clarify from my earlier post: I agree that is what their assumptions lead to, but I'm not convinced those assumptions are valid in the real world.
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Old 18th September 2009, 02:06 AM   #55
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Originally Posted by CoolSceptic View Post
The paper estimated that a total of 250,000 tosses would be needed to confidently identify a bias of the magnitude they describe.
How long before Derren uses this monumental bias as an explanation for a trick?
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Old 18th September 2009, 02:55 AM   #56
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Originally Posted by CoolSceptic View Post
Just to clarify from my earlier post: I agree that is what their assumptions lead to, but I'm not convinced those assumptions are valid in the real world.
The argument is really simple. Flip a coin that starts heads up and it spends more of its time in the air heads up, unless you flipped it exactly right (angular momentum perfectly parallel to the coin), in which case it's 50-50 heads-tails. It's impossible to flip it exactly right.

So, which assumption do you think might be invalid?

Last edited by sol invictus; 18th September 2009 at 02:56 AM.
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Old 18th September 2009, 03:02 AM   #57
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Originally Posted by sol invictus View Post
The argument is really simple. Flip a coin that starts heads up and it spends more of its time in the air heads up, unless you flipped it exactly right (angular momentum perfectly parallel to the coin), in which case it's 50-50 heads-tails. It's impossible to flip it exactly right.

So, which assumption do you think might be invalid?
Simple thought experiment for you. What do you think happens when there is a non-zero angular difference between the angle at which the coin is tossed and the angle at which it is caught?

Do you think that the angular difference will always be *exactly* zero, i.e. the catching angle will be exactly identical to the initial toss?
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Old 18th September 2009, 03:16 AM   #58
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Quote:
A good way of thinking about this is by looking at the ratio of odd numbers to even numbers when you start counting from 1.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Starting at 1 and ending at 17 is the bias.

Starting at 1 and ending at 18 would yield equal odd and even numbers.

If you were to do an infinite amount of coin tosses, there would be no way to determine if there's more odds and evens, and it would be 50/50.
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Old 18th September 2009, 03:35 AM   #59
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Originally Posted by CoolSceptic View Post
Simple thought experiment for you. What do you think happens when there is a non-zero angular difference between the angle at which the coin is tossed and the angle at which it is caught?

Do you think that the angular difference will always be *exactly* zero, i.e. the catching angle will be exactly identical to the initial toss?
That's one of the things that might matter, yes. Air resistance is another, as is catching time/time of flight. They comment on a list of such assumptions.

So, will you answer my question? Which of those do you think is important?

The one you raised doesn't sound it, because so long as the angle is less than 90 degrees their argument still works.
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Old 18th September 2009, 04:10 AM   #60
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I'd never trust a coin flip to be fair, unless I do it myself, in which case I know it's not fair.
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Old 18th September 2009, 04:25 AM   #61
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Originally Posted by Kahalachan View Post
Starting at 1 and ending at 17 is the bias.

Starting at 1 and ending at 18 would yield equal odd and even numbers.

If you were to do an infinite amount of coin tosses, there would be no way to determine if there's more odds and evens, and it would be 50/50.
The quote in the OP seems to be based on a misreading of the paper in a blog post. The argument in the paper is not based on the coin being heads up more times than not due to a simple counting argument, it's about it being heads up for longer than not due to angular momentum.
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Old 18th September 2009, 04:27 AM   #62
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Doesn't the findings of the real world tossing experiments indicate that their model, or their assumptions are flawed? A 2% bias shouldn't require 250,000 tosses to demonstrate should it?

Last edited by shuttlt; 18th September 2009 at 04:29 AM.
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Old 18th September 2009, 04:46 AM   #63
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Originally Posted by shuttlt View Post
Doesn't the findings of the real world tossing experiments indicate that their model, or their assumptions are flawed? A 2% bias shouldn't require 250,000 tosses to demonstrate should it?
It depends on the confidence level you want. 250,000 would give very high significance. But you'll need at least 40,000 or so to be reasonably certain.
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Old 18th September 2009, 05:38 AM   #64
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Originally Posted by sol invictus View Post
That's one of the things that might matter, yes. Air resistance is another, as is catching time/time of flight. They comment on a list of such assumptions.
Right. Which is why I said they were right for their assumptions, but I'm not convinced their assumptions are right.

Originally Posted by sol invictus View Post
So, will you answer my question? Which of those do you think is important?
? I've already answered this, the catch angle will not be the same as the toss angle, and the distributions they have assumed for this (I think) are unrealistic. This doesn't make them wrong, it just doesn't mean they've made their case.

Originally Posted by sol invictus View Post
The one you raised doesn't sound it, because so long as the angle is less than 90 degrees their argument still works.
Wrong, and they even acknowledge this in the paper. The bias can be reversed with angles of much less than 90 degrees. Think about the percentage of time the normal of the coin projected on to the great circle spends above the equator.
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Old 18th September 2009, 05:40 AM   #65
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Originally Posted by sol invictus View Post
It depends on the confidence level you want. 250,000 would give very high significance. But you'll need at least 40,000 or so to be reasonably certain.
Wow, that would take a while.
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Old 18th September 2009, 07:41 AM   #66
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Originally Posted by CoolSceptic View Post
Wrong, and they even acknowledge this in the paper.
Where do they acknowledge it? I missed that.

Quote:
The bias can be reversed with angles of much less than 90 degrees. Think about the percentage of time the normal of the coin projected on to the great circle spends above the equator.
I was wrong about 90 degrees, that's not the relevant angle. But I don't see how this changes their qualitative conclusions. Yes, if you always flipped the coin with some angular momentum and always caught it at the exact angle required to compensate, you could get a fair flip (I think). But if you want to be even remotely realistic, you should put a distribution on both those, and I think it's only going to be a set of measure zero of those distributions that give an unbiased answer.

Basically, what I'm saying is that the coin flip is inherently asymmetric, and so to compensate for that perfectly you have to do something very special. A generic flip and catch sequence won't do that.
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Old 18th September 2009, 08:47 AM   #67
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Originally Posted by sol invictus View Post
Where do they acknowledge it? I missed that.
First para, page 14 of the preprint linked in the OP. Here they discuss a generalisation in which they attempt to account for the difference in flipping and catching angles, and note that there is an angle at which the bias reverses. I wish they had chosen a different terminology for "same side" and "different side", but they refer to this as heads and tails respectively, so where they refer to a bias towards tails, they mean the bias switches to the other side of the coin.


Originally Posted by sol invictus View Post
Yes, if you always flipped the coin with some angular momentum and always caught it at the exact angle required to compensate, you could get a fair flip (I think). But if you want to be even remotely realistic, you should put a distribution on both those, and I think it's only going to be a set of measure zero of those distributions that give an unbiased answer.

Basically, what I'm saying is that the coin flip is inherently asymmetric, and so to compensate for that perfectly you have to do something very special. A generic flip and catch sequence won't do that.
They way they get around it in the paper is (IMHO) a bit tricky. They assume orthogonal, normally distributed components when consider a flip angle different to the catch angle. This results in an unbiased distribution of flips about the nominal catch angle.

I'm not convinced by this. I suspect the vast majority of coin flippers would have a bias in the angle between flip and catch. Just because you flip and catch naturally and the most natural position is unlikely to be centred on a zero angle.

Because the angle can result in a bias in the opposite sense (i.e. opposite side rather than same side), you can no longer conclude a consistent bias to one side between the two extremes. It might be the case that there is, but it doesn't automatically follow.
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Old 18th September 2009, 10:20 AM   #68
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Originally Posted by balrog666 View Post
A trained person can make it come up whatever side they want. Any practiced amateur can make a coin come up the same side with an extremely high probability.

Just takes practice.
The first post that makes sense to me. And that includes the OP. Scary
Thanks balrog665 for that sanity amongst the
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Old 18th September 2009, 10:25 AM   #69
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Originally Posted by arthwollipot View Post
I have found that I can usually predict what side the coin will land based on what side is up when I begin the toss. Not 100% reliably, but certainly better than chance alone.

Of course, I have never done any actual study of this.
Reading the rest made my head ache worse. Except for this beauty.
Excellent
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Old 18th September 2009, 11:38 AM   #70
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Originally Posted by Wowbagger View Post
He assumes a coin has a memory of how many times it was on one side versus another.

Most of the time it will never be quite exactly 50/50. But, which side gets slightly more has nothing to do with which side came up first.
Uh... it kinda does have a memory.

Namely, each time the coin undergoes a mechanical process of turning over, it has to 'remember' what side it was on on the last time it turned over.


This seems plausible, as it is handing you a piece of information about a random process, which gives you more information about the process.

Or, classically: A couple has two kids. If one is a boy, what are the odds the other one is a girl?

Information about a random process...
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Old 18th September 2009, 05:14 PM   #71
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Originally Posted by CoolSceptic View Post
Because the angle can result in a bias in the opposite sense (i.e. opposite side rather than same side), you can no longer conclude a consistent bias to one side between the two extremes. It might be the case that there is, but it doesn't automatically follow.
I agree that's possible. But what I pointed out in my last post is that for almost all such distributions of catch angle, there will still be a bias (and it will be of order the one they estimate, if their other assumptions are valid).
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Old 19th September 2009, 01:58 AM   #72
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Originally Posted by sol invictus View Post
I agree that's possible. But what I pointed out in my last post is that for almost all such distributions of catch angle, there will still be a bias (and it will be of order the one they estimate, if their other assumptions are valid).
It is certainly possible. But since a bias in the angle can cause the bias to reverse (to a bias of the reverse side), and since any spread in the distribution will encapsulate cases where the bias is both one side or the other, I suspect they have over estimated the bias, and it could be less than they expect.

That said, this is all conjecture, until the maths are worked through... I think step one would be to conduct an experiment to get a feel of the typical catch / toss angle bias and distribution. Do we have any experienced tossers on this forum?
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Old 19th September 2009, 06:54 AM   #73
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Originally Posted by !Kaggen View Post
The first post that makes sense to me. And that includes the OP. Scary
Thanks balrog665 for that sanity amongst the
Originally Posted by balrog666 View Post
A trained person can make it come up whatever side they want. Any practiced amateur can make a coin come up the same side with an extremely high probability.

Just takes practice.
Except that it's utter nonsense, unless balrog has simply decided to ignore my challenge rather than seeking to validate his claim:
Originally Posted by Southwind17 View Post
An investment of time, then, that Derren Brown, of all people, is not prepared to make, electing, instead, to spend around 9 hours "randomly" flipping a coin until heads shows 10 times in succession! How long would he need to practice for, do you think, to achieve "extremely high probability" of 10 in a row? I suspect you're hypothesizing here, and have never actually seen anybody demonstrate such a skill (assuming otherwise "fair" tossing of the coin, of course, by which I mean a high toss with fast spin rate).
Originally Posted by CoolSceptic View Post
That said, this is all conjecture, until the maths are worked through... I think step one would be to conduct an experiment to get a feel of the typical catch / toss angle bias and distribution. Do we have any experienced tossers on this forum?
And if we ever get beyond conjecture what then?
This is all academic, given that there are absolutely no conceivable scenarios in which any bias would have a real and practical effect.
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Old 19th September 2009, 12:23 PM   #74
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Originally Posted by Southwind17 View Post
And if we ever get beyond conjecture what then?
This is all academic, given that there are absolutely no conceivable scenarios in which any bias would have a real and practical effect.
Well... I can't see any immediate use of this research. But that doesn't mean it isn't an interesting puzzle to solve; not all puzzles need a reason for solving them.

That said, gaming statistics can be worth a lot of money. As it happens, I can't think of any casinos that use coin flipping as a standard game. But loading a game by as little as 1% in a casino can be worth a lot of money to somebody...
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Old 19th September 2009, 12:43 PM   #75
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Originally Posted by CoolSceptic View Post
... loading a game by as little as 1% in a casino can be worth a lot of money to somebody...
Assuming unbiased dice outcomes, the casino advantage for the Pass Line bet in craps is only $7 out of every $495 bet or about 1.414%. If dice can be thrown with a bias that overcomes that figure then playing craps can be profitable.
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Old 19th September 2009, 01:28 PM   #76
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Originally Posted by Towlie View Post
Assuming unbiased dice outcomes, the casino advantage for the Pass Line bet in craps is only $7 out of every $495 bet or about 1.414%. If dice can be thrown with a bias that overcomes that figure then playing craps can be profitable.
In principle a similar argument could work for dice, although the dynamics when they hit the table are much more complicated. Still, that's interesting.

But bear in mind that reversing the advantage of the casino is actually not enough to be able to earn money. The casino has much more money than you do (I assume), which means the situation is asymmetric and you'd need to place many of small bets to leverage your advantage.

With a 1% edge you'd need lots of time playing craps to earn anything (same reason why this coin bias requires so many flips to test).
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Old 19th September 2009, 02:12 PM   #77
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Originally Posted by sol invictus View Post
But bear in mind that reversing the advantage of the casino is actually not enough to be able to earn money. The casino has much more money than you do (I assume), which means the situation is asymmetric and you'd need to place many of small bets to leverage your advantage.

With a 1% edge you'd need lots of time playing craps to earn anything (same reason why this coin bias requires so many flips to test).
In that regard it's no different than card counting. The optimum strategy would be to size your bets according the Kelly CriterionWP and play as much as possible.
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Old 19th September 2009, 07:13 PM   #78
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Originally Posted by Towlie View Post
In that regard it's no different than card counting. The optimum strategy would be to size your bets according the Kelly CriterionWP and play as much as possible.
Yes, exactly.

With only a 1% edge it's generally more work than it's worth, unless you have a very large bankroll.
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Old 19th September 2009, 11:55 PM   #79
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Originally Posted by sol invictus View Post
But bear in mind that reversing the advantage of the casino is actually not enough to be able to earn money. The casino has much more money than you do (I assume), which means the situation is asymmetric and you'd need to place many of small bets to leverage your advantage.

With a 1% edge you'd need lots of time playing craps to earn anything (same reason why this coin bias requires so many flips to test).
Seems you're alluding to the stereotypical professional gambler here!

Originally Posted by sol invictus View Post
With only a 1% edge it's generally more work than it's worth, unless you have a very large bankroll.
Ditto, but remember that a very large bankroll will soon come up against the house limits, and in any event a clearly winning strategy will very quickly result in the casino manager politely showing you the exit sign, via the restaurant, of course!
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Old 20th September 2009, 07:24 AM   #80
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There's a lot of misunderstanding about positive expectation gambling here. Maybe we should get a thread going about the subject.
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