

Coin toss not random after all - anigbrowl
http://www.mercurynews.com/top-stories/ci_13579962?nclick_check=1

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shaddi
If you've taken a basic probability class, then you probably know the solution
to this issue of obtaining a fair toss with a biased coin. Flip the coin
twice. If both are heads, or both are tails, discard the pair of tosses and
try again. If the first flip is a heads and the second is a tails, then count
the pair as a "heads". If the first is a tails and the second is heads, count
the pair as "tails". Even with a bias, the probability of getting a heads-tail
pair equals the probability of getting a tails-heads pair.

Here's a blog post that explains this in more detail:
[http://www.billthelizard.com/2009/09/getting-fair-toss-
from-...](http://www.billthelizard.com/2009/09/getting-fair-toss-from-biased-
coin.html)

You can be sure all future coin tosses I do will use this technique!

~~~
ars
It's not a biased coin.

It's a biased flip.

Your solution does not help with that.

~~~
sorbus
Then simply have the same side facing up each time; in that case, there would
be no difference between a coin which is biased and a flip which is biased -
the bias remains in the same direction, and thus the method works.

~~~
shaddih
(I noprocrast'd myself out, but I couldn't stop thinking about this.)

It's still an issue, GP is very right. The problem is that you can get any
desired outcome with some probability p: it's effectively a coin that changes
bias depending on the flipper's preference.

I'm trying to think of a solution that resolves this... thoughts encouraged.
:)

~~~
bh3
Only thing that comes to mind is to add some sort of blind element. For
example having two people, one flips two coins but both remain ignorant of the
actual outcome and assume that they have opposing interests (and so would not
be likely to try to help the other in any way). Depending on the outcome of
the preliminary flip, the one of the second person is either kept the same or
inversed. Again still not perfect but removes some level of manipulation.
Which is sort of interesting because you'd have to wonder should the
preliminary tosser try to make the probability as even or skewed as possible.
If he is likely to skew it it becomes a whole new scenario.

~~~
nfnaaron
Since the bias is in the toss, not the coin:

Put the coin in the ref's hands, which are cupped against each other in a
closed sphere. The ref shakes his hands to the satisfaction of all (and we
assume an honest ref). He then places the coin on his thumb, but uses the
opposite hand to cover the placing operation and the final launch
configuration.

After the coin is positioned, but before it's uncovered, one of the interested
parties calls it. Then the coin is tossed.

Fair, random, and evenly distributed as far as I can tell.

Edit: I think this works as well with two interested parties and an impartial
ref, or only two opponents where the non-tosser calls the coin before it's
tossed.

~~~
kelnos
In the case of an honest 3rd party doing the toss, you can even have someone
call it _before_ the ref shakes the coin around in his cupped hands. That way
you eliminate the possibility of someone seeing the coin's orientation as it's
placed on the thumb. The ref doesn't even need to hide the starting position
after that.

If one of the interested parties is also the coin tosser, this could offer an
opportunity to game the system, though, so one might want to just use your
proposal in all cases to avoid complexity.

Thinking about this a bit more, though, I wonder if this really has the
desired effect.

Let's assume for a moment that the act of the ref shaking the coin around in
his cupped hands actually generates a random starting position. Ok, so
starting position is 50/50 heads/tails, and the caller of the coin doesn't
know the starting position.

In that case, the final heads/tails probability then depends on the initial
starting position -- to use the bad-case numbers from the article, say the
final probability is 60/40, favoring the side that was initially facing up.

So what have we really gained here? All we've done is made sure the initial
positioning of the coin is random. The bias in the toss itself is still there,
but we're hiding it by making sure the caller doesn't know the initial state
of the coin.

If that's the case, then why not:

a) Don't even toss the coin at all. If we're convinced that the act of the ref
shaking the coin around in his cupped hands gives a random starting position,
then why not use that in place of the coin toss?

... or...

b) Avoid the whole cupped-hands shake thing entirely. Just have the interested
party call the toss before the ref places the coin on his thumb. Obviously
then you have to have an impartial ref who won't then place the coin on his
thumb in a way to benefit one of the interested parties. (You could just
specify that the ref is required to reach into his pocket, pull out the coin,
and place it on his thumb, all without looking at it at all or feeling its
surface sufficiently to figure out which side is which.)

I guess it also depends on what we care about to make this "random."
Personally I think it's random enough if the caller simply doesn't know the
starting position when calling the coin, assuming the coin tosser doesn't use
the call to game the toss.

Heh, or we can just admit that tossing a coin isn't sufficiently random, and
use something else... like radioactive isotope decay... or even a PC's PRNG
(though that of course opens a big computer security debate).

~~~
nfnaaron
"So what have we really gained here? All we've done is made sure the initial
positioning of the coin is random. The bias in the toss itself is still there,
but we're hiding it by making sure the caller doesn't know the initial state
of the coin."

Assuming the cupped hands shake produces a random starting position, you've
evenly distributed the toss bias, so the result of a series of tosses should
be evenly distributed.

"If that's the case, then why not: a) Don't even toss the coin at all. If
we're convinced that the act of the ref shaking the coin around in his cupped
hands gives a random starting position, then why not use that in place of the
coin toss?"

That's an excellent simplification, but it just doesn't feel as dramatic and
traditional to decide on shaking cupped hands. You need the toss for effect.

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scott_s
The claim is that it's not _uniformly_ random - all outcomes don't have equal
probability. But it's still a random process.

~~~
Flankk
Two coins flipped at equal velocity and trajectory in vacuum will flip an
equal number of times. Thus, the process of a coin flip is completely
deterministic since it depends on newtonian physics. So, the probability is
defined by the randomness of the _conditions_ , not the process itself.

When the conditions are controlled then so is the probability. The results of
this research should not be surprising despite the tone of the article.

~~~
mitko
You also could have air motions, electric fields and so on passing through the
trajectories of the coins. Unless you account for everything on atomic level,
you have all that uncertainty as random noise in you measured outcome. So
indeed it _is_ a random variable. Biased, yes, but random.

Actually, even if you can have data at atomic level you still have chaos
theory and butterfly effect preventing you from making prediction. Then even
some small fluctuation born by the Plank's "uncertainty principle" can grow up
in time to big enough error.

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mattmaroon
This is still random. Random just means the outcome is unpredictable, not that
all outcomes are equally likely. If a coin flip is 60% to be heads, 40% to be
tails, the results are still random.

Or, looked at another way, it's not random at all. It's a very simple function
of the coin's initial state and the forces applied to it. If you knew all of
those, you would be able to predict the results ahead of time.

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sachinag
Link fails for me - it goes to a registration check page or some such?

~~~
chaosmachine
try this?

[http://www.mercurynews.com/search/ci_13579962?nclick_check=1...](http://www.mercurynews.com/search/ci_13579962?nclick_check=1&forced=true)

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anigbrowl
and the paper in question: <http://comptop.stanford.edu/preprints/heads.pdf>

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danw
Check out "CoinFlipper" [0] - a series of machines designed to flip coins to
land on the side you specify. They're so brilliantly delightful. After months
of testing, experimenting and usage the coin only once landed on it's edge
with the machines.

[0] [http://www.dotmancando.info/index.php?/projects/coin-
flipper...](http://www.dotmancando.info/index.php?/projects/coin-flipper/)

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alan-crowe
Jaynes discusses this in Chapter 10 of Probability Theory: The Logic of
Science. His real aim is to seperate the concepts of having to take a decision
without knowing all the facts versus having to take a decision in the face of
some kind of intrinsic randomness. Having seperated the two concepts he then
argues that probability theory is about the former, with the later something
of a red herring.

So he emphasises that coin tosses obey the laws of physics and are not random,
but we use them anyway because we don't know the facts that we need to know in
order to call the toss. He included experimental results of successful
cheating, but with a jam jar lid instead of a coin. Clearly we call a coin
toss "random" because of the practical difficulties involved in predicting the
motion of a disk much smaller than a jam jar lid.

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henning
Those football players who reject the paper out of hand without considering
its merits are the ones who are "ridiculous". One of the authors is Persi
Diaconis. If he says something about statistics, you don't get to just reject
it.

~~~
kingkawn
yeah, what did those football players do their entire college careers, play
games?

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rjprins
This can also be theoretically proven:

If we view a coin flip as a random variable X that describes the number of
revolutions, then X has a value of 0 to infinity.

Before the coin is flipped, lets say H faces up. Then as X marks the number of
revolutions, R will describe the result:

    
    
      X: 0 1 2 3 4 5 6 ..
      R: H T H T H T H ..
    

As you can see for any value of X, the number of times H was up is always
equal or _once more_ then T. This shows a clear bias for the side facing up.

~~~
pmjordan
Uhhh, careful with "once more" in the context of infinity.

The way to model this would be to come up with a probability distribution for
the number of revolutions. If the distribution is skewed towards few
revolutions, (something like a poisson distribution, say) then it's very
likely that the outcomes have probabilities ≠50% due to the discretisation.

In practical terms, I guess the person flipping the coin should be required to
flip it such that it rotates very fast, which ought to provide a gaussian
distribution around a high number of revolutions.

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JCThoughtscream
Anybody who's deliberately "fumbled" a coin toss before knows that it's all in
the flick. Especially if you were to, say, put more emphasis on the side of a
coin, giving it a lopsided spin so as to make it /look/ as if it's flipping at
a casual glance.

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schemer
Craps players have long known about controlling the outcome of a dice toss.

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mitko
Any statistical significance results? Without them I am not buying it.

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Syama
btw Boston was not the loser in the Portland Oregon coin toss, it was
Stumptown.

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icefox
man I am glad the football is the "important" example they gave.

~~~
nfnaaron
I thought it was an entertaining and effective way to show something
mathematical to the general public, in a way that most people can relate to.

I was especially tickled near the end of the article where they described
discussions on changing the overtime rules. When they mentioned the players'
reaction, I thought it would turn on something related to winning, and I was
surprised to learn that their objection was more playing time and more chance
of injury.

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RK
Coin spin instead?

~~~
gjm11
Spinning coins tends to introduce considerably more bias than tossing them.

