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Old 15th September 2009, 08:46 AM   #1
Towlie
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Coin Flipping Probabilities - Does this sound right to you?

I just now came upon a reference in the latest issue of Crypto-Gram to this article by a blogger named James Devlin. It's an interesting discussion of a paper titled Dynamical Bias in the Coin Toss.

There's one particular point in this article that bothers me and that I'd like to discuss here. I haven't yet looked at the paper itself, but James Devlin says this, parts of which may or may not be supported by the paper:

Quote:
Here are the broad strokes of their research:

1. If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there's a 51% chance it will end as heads).

(Premises 2 - 7 omitted.)

The 51% figure in Premise 1 is a bit curious and, when I first saw it, I assumed it was a minor bias introduced by the fact that the "heads" side of the coin has more decoration than the "tails" side, making it heavier. But it turns out that this sort of imbalance has virtually no effect unless you spin the coin on its edge, in which case you'll see a huge bias. The reason a typical coin toss is 51-49 and not 50-50 has nothing to do with the asymmetry of the coin and everything to do with the aggregate amount of time the coin spends in each state, as it flips through space.

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

No matter how long you count, you'll find that at any given point, one of two things will be true:

* You've touched more odd numbers than even numbers
* You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

* You've touched more even numbers than odd numbers.

Similarly, consider a coin, launched in the "heads" position, flipping heads over tails through the ether:

H T H T H T H T H T H T H T H T H T H T H T H T H

At any given point in time, either the coin will have spent equal time in the Heads and Tails states, or it will have spent more time in the Heads state. In the aggregate, it's slightly more likely that the coin shows Heads at a given point in time—including whatever time the coin is caught. And vice-versa if you start the coin-flip from the Tails position.
Does that sound right to you? If there is such a bias, is that a correct explanation for it?

Most people catch a flipped coin and then slap it down on the back of their wrist. In that case, does that mean that there's a 51% chance of the coin landing on the opposite face of the one it was launched from?

When I get time, I'd like to try writing a computer simulation to test what James Devlin says. Meanwhile, what do you folks think about this?
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Old 15th September 2009, 09:09 AM   #2
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The solution, obviously, for a fair toss, is to snatch the coin out of the air with a sideways grab before slapping onto the back of the other hand.
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Old 15th September 2009, 09:39 AM   #3
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Originally Posted by Towlie View Post
Does that sound right to you? If there is such a bias, is that a correct explanation for it?
That's not a bias; the quantity you're finding a bias in ("the number of times the coin passes through its initial state, over its whole trajectory") is not the same as the number you actually measure in a coin flip ("the state at some random time T after the flip when I grab/catch the coin").

You can make those probabilites agree if you flip the coin in a very unusual way: by choosing T in an entirely flat distribution between T=0 (coin hasn't flipped yet) and some later time. That's not what you do in a real coin toss; I suspect that you grab in some window between T=1 second and T=1.5 seconds, or something, and there's no clear bias in that statistic.

Look at it another way: suppose that, rather than counting from 1, you count from 2---that's like letting the coin take one full flip before allowing yourself to even think about catching it. Now there's a bias towards even numbers, of the same size as the previous "bias" towards odd numbers---based on the apparently-meaningless decision not to catch the coin at a moment when you would never have caught the coin anyway. That should tell you that neither bias is really there.
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Old 15th September 2009, 11:54 AM   #4
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Can't have a thread about coin flipping without a reference to this guy:


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Old 15th September 2009, 12:01 PM   #5
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As long as you remember that the coin has no memory

You won't be taken by some Maverick-looking sharp:

http://www.scribd.com/doc/13366448/T...lay-in-One-Act
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Old 15th September 2009, 01:38 PM   #6
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Originally Posted by Towlie View Post
Does that sound right to you? If there is such a bias, is that a correct explanation for it?
It doesn't sound right to me. I agree with ben m's explanation of why not, which occurred to me as well, even before I read his post.

Originally Posted by Towlie View Post
Most people catch a flipped coin and then slap it down on the back of their wrist. In that case, does that mean that there's a 51% chance of the coin landing on the opposite face of the one it was launched from?
Sure. If one side has, for whatever reason, a 51% chance, the other side must have a 49% chance.

Originally Posted by Towlie View Post
When I get time, I'd like to try writing a computer simulation to test what James Devlin says. Meanwhile, what do you folks think about this?
I think that a simulation of lots of tosses, using pseudo-random numbers, isn't very efficient. You'll need to choose a probability distribution for when the coin lands (i.e., how likely is it to land after half a twist in the air, how likely after one full twist, how likely after a twist and a half, etc.), and once you have that, you can compute the exact probability of heads without simulating any tosses at all -- just add up all the probabilities for integer numbers of twists (or integer + 1/2, depending on whether heads or tails faced up to begin with).
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Old 15th September 2009, 01:52 PM   #7
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I am assuming that the author is not saying that a coin has 'memory' - it sounds like he's saying that the mechanical process of flipping contains a slight bias toward repeating the previous outcome.

Sounds plausible.

I doubt it's related to differential weight of the coin.

Sort of like knife throwing: throw a knife ten times, and it'll hit the wall the same way because that's the way you're throwing it. The spin is more consistent than it is randomizing.
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Old 15th September 2009, 02:05 PM   #8
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I assume it is the consistency of tossing AND consistency of catching.

Seems to me from my back-yard knife throwing, that each knife has a set rotational speed. Has to do with 'balance'. So, actually, there ought to be a vary plausible way for someone to learn to toss a coin and get up-face lots more than 50%. Just like a knife thrower gets the knife to stick very regularly. The way I threw, I believe it mad 1/2 turn every two feet. So if I stepped back two feet, I would reverse the knife in my hand. You would need to learn that for a coin. Same 'toss' to same height, same catch. Somebody with exceptional dexterity ought to be able to improve on 50%.... start by building a machine, just to prove the concept ?
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Old 15th September 2009, 02:13 PM   #9
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The physics of the coin toss and a heuristic for why most coin tosses are biased is give on pages 3 and 4 of the pdf of the original article.
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Old 15th September 2009, 02:18 PM   #10
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If you're worried about generating good random numbers from a biased coin toss, use the following procedure: toss a coin *twice*. There are four possible results: HH, TT, HT, and TH. Then follow these rules:

If it's HH, repeat the two tosses.
If it's TT, repeat the two tosses.
If it's HT, interpret that as a "win"
If it's TH, interpret that as a "loss".

That win/lose distribution should be 50-50, regardless of how unfairly weighted the coin is. (The worst that can happen, if the raw coin flips are, say, 90% H 10% T, is that you waste a lot of time discarding and repeating multiple HH results.)

I forget what this is called, but I thought it was neat when I first learned about it.
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Old 15th September 2009, 02:26 PM   #11
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Absolute nonsense.

Consider a 3 sided coin. Use the same logic in the blog:

When you start the flip, it is on face 1. Then 2. Then 3.

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

The only time that 1,2, and 3 have the same odds is when the 3rd face is up. Otherwise, their count is higher. Thus faces 1 and 2 should have higher odds than the third face. Furthermore, face 1 has an advantage over face 2, so it's odds will be higher. Thus 1 will show up most often, then 2, then finally 3.

Sorry, it just doesn't work that way.

edit: if this is not clear, consider this alternative method of computing the odds:

You flip a coin and catch it. The coin flips N times before being caught, where N is a random number. If N is even, the coin lands on the same face that started, if N is odd, it lands on the opposite face. Since N is random, odd and even is evenly distributed. (we could quibble that you never catch it on flip one or two, but that doesn't change the math. It's still random even if the lower limit is greater than one).
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Old 15th September 2009, 02:48 PM   #12
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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.
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Old 15th September 2009, 02:50 PM   #13
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Now, I only basing this my failing memory and I leave it to someone else to do the research but it strikes me from the OP that if the distance the coin is falling is constant there might just be something to the claim.

Some years ago ISTR there was an article in Scientific American about the myth (theory) that a slice of bread alway falls jam side down. The research did show that by varying the distance it fell the slice would preferentially land jam-side-down or jam-side-up. I forget whether there was some sort of "natural frequency of rotation" involved.

Maybe, under certain conditions, this would apply to coins as well?

YMMV & etc.
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Old 15th September 2009, 03:22 PM   #14
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Originally Posted by Gord_in_Toronto View Post
Now, I only basing this my failing memory and I leave it to someone else to do the research but it strikes me from the OP that if the distance the coin is falling is constant there might just be something to the claim.

Some years ago ISTR there was an article in Scientific American about the myth (theory) that a slice of bread alway falls jam side down. The research did show that by varying the distance it fell the slice would preferentially land jam-side-down or jam-side-up. I forget whether there was some sort of "natural frequency of rotation" involved.

Maybe, under certain conditions, this would apply to coins as well?

YMMV & etc.
But this assumes that the amount of time bread takes to hit the floor is consistent and approximately the same as the amount of time it takes for the bread to turn 180 degrees. In the case of the coin it turns many times before it lands and both the rate of turn and the height the coin drops varies. Unless the initial impulse, height, etc is very consistent there would be no bias. The bias could be to either heads or tails depending on minor differences in the initial conditions.
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Old 15th September 2009, 03:50 PM   #15
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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.
That's not what he's doing at all.
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Old 15th September 2009, 04:36 PM   #16
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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 15th September 2009, 05:27 PM   #17
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Originally Posted by shuttlt View Post
But this assumes that the amount of time bread takes to hit the floor is consistent and approximately the same as the amount of time it takes for the bread to turn 180 degrees. In the case of the coin it turns many times before it lands and both the rate of turn and the height the coin drops varies. Unless the initial impulse, height, etc is very consistent there would be no bias. The bias could be to either heads or tails depending on minor differences in the initial conditions.
One could argue (not that I would of course) that the very appearance of the words "assumes", "consistent", "approximately", "varies" and "minor differences" could very well result in a small bias of 51% in the results. In the case of the jammed bread, with all factors under control the bias was, ISTR, 100%.
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Old 15th September 2009, 07:20 PM   #18
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Originally Posted by Gord_in_Toronto View Post
One could argue (not that I would of course) that the very appearance of the words "assumes", "consistent", "approximately", "varies" and "minor differences" could very well result in a small bias of 51% in the results. In the case of the jammed bread, with all factors under control the bias was, ISTR, 100%.
That could easily be tested by flipping a blank with different colored sides.
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Old 15th September 2009, 09:41 PM   #19
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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.
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Old 15th September 2009, 09:57 PM   #20
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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.
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Old 15th September 2009, 10:18 PM   #21
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A problem with the series HTHTHTHT..... being biased towards H is that, when applied to the coin toss, it assumes that the coin is equally likely to land on any value in the series. I doubt that this is the case. If it's more likely to land on the second value than the first, and the third value more the the second, it stops mattering half so much how many heads and tails are in the series.
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Old 15th September 2009, 10:57 PM   #22
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ben m and shuttit have the right answer here...

The argument in the OP assumes that the probability distribution for flips is uniform (or at least some distribution which is either constant or monotonically decreasing, e.g. exponential). In this case, the bias they present is a valid one because the probability of a single flip is greater than two flips is greater than three flips etc.

In practice the number of flips is almost certainly not like this. (Of course, it depends a little bit on the person doing the flipping - their flipping style, so to speak, will influence the probabilities).

Most likely the probability distribution of the number of flips will be different - some non-negative shaped distribution (don't know which; binomial perhaps, could be poisson, rayleigh, chi-square etc). The overall probability is then the sum of the probability of N flips times the outcome (from 0 to inf).

The distribution (which is almost certainly a function of the flipper) will dictate whether a bias is present, but if the PDF has any significant shape to it (likely) then it does not automatically follow that a bias is present.
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Old 15th September 2009, 11:31 PM   #23
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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

No matter how long you count, you'll find that at any given point, one of two things will be true:

* You've touched more odd numbers than even numbers
* You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

* You've touched more even numbers than odd numbers.

For this to be true, it assumes that a valid flip includes 'flipping' the coin ZERO times as well as any amount up to infinity?

Flipping the coin zero times is course an invalid flip. It's not flipping the coin at all, it's just leaving the coin in its initial state, so the list should read: 2,3,4,5,6,7,8,9,etc..

50/50
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Old 15th September 2009, 11:42 PM   #24
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Originally Posted by kevinquinnyo View Post
For this to be true, it assumes that a valid flip includes 'flipping' the coin ZERO times as well as any amount up to infinity?

Flipping the coin zero times is course an invalid flip. It's not flipping the coin at all, it's just leaving the coin in its initial state, so the list should read: 2,3,4,5,6,7,8,9,etc..

50/50
According to the argument in the OP, this would then put the bias in favor of even numbers as the number of evens is always equal to, or one greater than the number of odds.
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Old 16th September 2009, 01:54 AM   #25
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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

No matter how long you count, you'll find that at any given point, one of two things will be true:

* You've touched more odd numbers than even numbers
* You've touched an equal amount of odd numbers and even numbers

What will never happen, is this:

* You've touched more even numbers than odd numbers.

Similarly, consider a coin, launched in the "heads" position, flipping heads over tails through the ether:

H T H T H T H T H T H T H T H T H T H T H T H T H

At any given point in time, either the coin will have spent equal time in the Heads and Tails states, or it will have spent more time in the Heads state. In the aggregate, it's slightly more likely that the coin shows Heads at a given point in time—including whatever time the coin is caught. And vice-versa if you start the coin-flip from the Tails position.

I don't think this has any relevance... partly because this isn't about "heads" and "tails" but "same" and "different", but mostly because any number of coin-flips necessary to test this with stastistical signifigance will result in this effect being a very tiny fraction of a percent that gets lost when the final result is rounded off to a convenient number of signifigant digits.
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Old 16th September 2009, 02:14 AM   #26
sol invictus
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Originally Posted by kevinquinnyo View Post
For this to be true, it assumes that a valid flip includes 'flipping' the coin ZERO times as well as any amount up to infinity?

Flipping the coin zero times is course an invalid flip.
"Invalid" according to who?

As is pointed out in the paper, it's possible to "flip" a coin with 0 flips in a way that's visually indistinguishable from a normal flip. You just give it the right angular momentum, so it preceses but doesn't actually flip over.
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Old 16th September 2009, 03:18 AM   #27
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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.
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 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.
And I doubt you're otherwise tossing the coin "fairly" either (see above).

Originally Posted by shuttlt View Post
A problem with the series HTHTHTHT..... being biased towards H is that, when applied to the coin toss, it assumes that the coin is equally likely to land on any value in the series. I doubt that this is the case. If it's more likely to land on the second value than the first, and the third value more the the second, it stops mattering half so much how many heads and tails are in the series.
Re. throw-away emboldening, on what basis is your doubt founded?

Coming back to Derren Brown's coin toss stunt I now find something particularly interesting: his selection of heads over tails. Why? Two reasons:
  1. Well, it's hard to be sure, but looking at the patterns I'd say that the heads side of a UK 10 pence coin contains slightly more alloy than the tails side, meaning that if there were a bias based on weight distribution the odds would be weighted against 10 heads occurring before 10 tails.
  2. Why didn't he cleverly present and film the stunt such that either 10 heads or 10 tails would have sufficed, thereby potentially saving a lot of laborious filming time? Or did he?!?
That stated, if there were a 49/51% weight bias, unless my maths ability fails me (probably!) that equates to a one in a hunderd million trillion probability that such bias would account for each of 10 successive outcomes. Hardly worth DB taking into account, given the other counteracting variables, such as starting position, toss height, rotation speed, toss direction, point of impact with glass bowl, etc.
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Old 16th September 2009, 03:50 AM   #28
shuttlt
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Originally Posted by Southwind17 View Post
Re. throw-away emboldening, on what basis is your doubt founded?
Why would one expect an even distribution?

Is the odds of the coin flipping a billion times the same as the odds of it flipping twice? If not, then either we have a bunch of identical probability outcomes that suddenly drop to zero at, say 30 flips, or we have non-identical probability numbers of flips that trail off to zero as the number of flips increases
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Old 16th September 2009, 04:02 AM   #29
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Originally Posted by Southwind17 View Post
That stated, if there were a 49/51% weight bias, unless my maths ability fails me (probably!) that equates to a one in a hunderd million trillion probability that such bias would account for each of 10 successive outcomes. Hardly worth DB taking into account, given the other counteracting variables, such as starting position, toss height, rotation speed, toss direction, point of impact with glass bowl, etc.
While I don't this a 49/51 bias is worth worrying about in this context, I think it's over the top to ask that the bias be responsible for each of the ten successful flips. We're expecting 5 to be successful by chance after all.

Odds of ten heads in a row with 50/50 odds: 0.00098
Odds of ten heads in a row with 51/49 odds: 0.00119

Assuming you kept tossing the coin in complete runs of ten (which I'm sure Derren didn't), I make it as taking 706 runs to reach 50% odds off success with a fair coin, but only 582 runs with our 51/49 coin. Perhaps someone would like to check my maths, basically I've just done:

(required number of runs)=log(0.5)/log(1-0.00098)
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Old 16th September 2009, 04:12 AM   #30
Southwind17
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Originally Posted by shuttlt View Post
Why would one expect an even distribution?
Er ... because the odds are 50/50?!

Originally Posted by shuttlt View Post
Is the odds of the coin flipping a billion times the same as the odds of it flipping twice?


Originally Posted by shuttlt View Post
If not, then either we have a bunch of identical probability outcomes that suddenly drop to zero at, say 30 flips, or we have non-identical probability numbers of flips that trail off to zero as the number of flips increases
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Old 16th September 2009, 04:20 AM   #31
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Originally Posted by shuttlt View Post
We're expecting 5 to be successful by chance after all.
?

Originally Posted by shuttlt View Post
Odds of ten heads in a row with 50/50 odds: 0.00098
Odds of ten heads in a row with 51/49 odds: 0.00119
Correct, but only +1/-1 of the 51/49 probability is attributable to the bias, i.e. 2%
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Old 16th September 2009, 04:23 AM   #32
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Originally Posted by Southwind17 View Post
Er ... because the odds are 50/50?!





No need to be rude.

Look, in the OP the claim is made that the 51/49 bias is to do with the series of values HTHTHT.... that the coin makes in the air:
Quote:
The reason a typical coin toss is 51-49 and not 50-50 has nothing to do with the asymmetry of the coin and everything to do with the aggregate amount of time the coin spends in each state, as it flips through space.
It is argued that because the number of times that the coin has been heads as it flips (assuming that we start with heads up) will always be equal to, or greater, than the number of time that the coin has been tails, then the odds of it coming up heads are greater.

The problem is that this argument implicitly assumes that the odds of each flip being the final one are identical for all flip counts. This is clearly not the case under real conditions.

I'm quite prepared to believe that there is a slight bias to heads caused by the asymmetry in the coin.
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Old 16th September 2009, 04:29 AM   #33
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Originally Posted by Southwind17 View Post
?
if I say I'm going to toss a coin 10 times and produce 10 heads, you expect me to get 5 by chance and probably aren't too surprised if I get 7.

Originally Posted by Southwind17 View Post
Correct, but only +1/-1 of the 51/49 probability is attributable to the bias, i.e. 2%
Do we agree then that if you were going to stand flipping a coin until you got 10 heads, it probably would save you a bunch of time if your coin had a bias as slight as 51/49 in favor of the face you were trying for?

Anyway, isn't it 0.02%?
(0.00119 - 0.00098)*100=0.021
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Old 16th September 2009, 04:48 AM   #34
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Originally Posted by shuttlt View Post
if I say I'm going to toss a coin 10 times and produce 10 heads, you expect me to get 5 by chance and probably aren't too surprised if I get 7.
Sorry - missed your point.

Originally Posted by shuttlt View Post
Do we agree then that if you were going to stand flipping a coin until you got 10 heads, it probably would save you a bunch of time if your coin had a bias as slight as 51/49 in favor of the face you were trying for?
Only if you could somehow take advantage of such bias, which you couldn't, given all of the other infinitely more influential factors (assuming an otherwise fair toss).

Originally Posted by shuttlt View Post
Anyway, isn't it 0.02%?
(0.00119 - 0.00098)*100=0.021
I was working on 1 percentage point (51% - 50%) being 2% of 50%.
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Old 16th September 2009, 05:00 AM   #35
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Originally Posted by Southwind17 View Post
Only if you could somehow take advantage of such bias, which you couldn't, given all of the other infinitely more influential factors (assuming an otherwise fair toss).
Say I have a weighted coin that gives a slight bias to heads owing to the location of the center of gravity of the coin. Theoretically the coin should spend 51% of it's time heads up if I position it just right when I toss it. Are you saying that in fact the motion of the coin is so swamped by random influences that if I tossed it from now until doomsday, the bias would not be apparent? In other words, there is no such thing as a biased coin?

Originally Posted by Southwind17 View Post
Sorry - missed your point.
Still, or do you now understand what I'm on about?

Originally Posted by Southwind17 View Post
I was working on 1 percentage point (51% - 50%) being 2% of 50%.
OK
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Old 16th September 2009, 11:37 AM   #36
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Originally Posted by shuttlt View Post
Say I have a weighted coin that gives a slight bias to heads owing to the location of the center of gravity of the coin. Theoretically the coin should spend 51% of it's time heads up if I position it just right when I toss it. Are you saying that in fact the motion of the coin is so swamped by random influences that if I tossed it from now until doomsday, the bias would not be apparent?
"Position it just right"! Clearly, you're suggesting an unfair toss, in ALL respects, in which case anything's possible. Otherwise, it depends, of course, on when your doomsday is. If it's around the same time as the army of monkeys are scheduled to finish the Complete Works of Shakespeare I can't argue with you. Otherwise, forget it for absolutely all intents and purposes.

Originally Posted by shuttlt View Post
In other words, there is no such thing as a biased coin?
Don't doubt the notion of a biased coin. In fact, all coins are biased, to some degree. Does it affect the likely outcome of a fair toss? Absolutely not.

Originally Posted by shuttlt View Post
Still, or do you now understand what I'm on about?
Understood
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Old 16th September 2009, 11:56 AM   #37
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Originally Posted by Southwind17 View Post
"Position it just right"! Clearly, you're suggesting an unfair toss, in ALL respects, in which case anything's possible. Otherwise, it depends, of course, on when your doomsday is. If it's around the same time as the army of monkeys are scheduled to finish the Complete Works of Shakespeare I can't argue with you. Otherwise, forget it for absolutely all intents and purposes.
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.

Originally Posted by Southwind17 View Post
"Don't doubt the notion of a biased coin. In fact, all coins are biased, to some degree. Does it affect the likely outcome of a fair toss? Absolutely not.
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 did find this quote:
"However, it is not possible to bias a coin fip—that is,
one cannot, for example, weight a coin so that it is substantially
more likely to land “heads” than “tails” when ¸ flipped and caught
in the hand in the usual manner."
http://www.stat.columbia.edu/~gelman...d/diceRev2.pdf

According to this it can be substantially biased if allowed to spin, or bounce...so maybe if you flipped it onto a table, or into a bowl. Anyway, it seems to me that this might still allow for a 51/49 bias, or maybe a 501/499 bias even if we define flipping to minimize the effect of bias as in the OP. It kind of depends on the word 'substantial'.

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.

Last edited by shuttlt; 16th September 2009 at 11:57 AM.
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Old 16th September 2009, 07:41 PM   #38
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Originally Posted by Southwind17 View Post
And I doubt you're otherwise tossing the coin "fairly" either (see above).
I think the fact that I get skewed, predictable results pretty much guarantees that I'm not tossing the coin fairly.
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Old 16th September 2009, 09:29 PM   #39
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Quote:
1. If the coin is tossed and caught, it has about a 51% chance of landing on the same face it was launched. (If it starts out as heads, there's a 51% chance it will end as heads).
The use of "heads" in this quote was clearly intended as just an example. From the first sentence it seems clear that they mean is...

If it starts on heads it has a 51% chance of ending up on heads.
If it starts on tails it has a 51% chance of ending up on tails.

So why are we talking about coins being slightly biased to show up one side more often than the other? That line of discussion seems irrelevant to the topic.
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Old 16th September 2009, 09:43 PM   #40
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Originally Posted by Brian-M View Post
The use of "heads" in this quote was clearly intended as just an example. From the first sentence it seems clear that they mean is...

If it starts on heads it has a 51% chance of ending up on heads.
If it starts on tails it has a 51% chance of ending up on tails.

So why are we talking about coins being slightly biased to show up one side more often than the other? That line of discussion seems irrelevant to the topic.
In the spinning coin trials, the bias was actually towards heads.
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