
The Coin Flip: A Fundamentally Unfair Proposition? (2009) - scandox
https://econ.ucsb.edu/~doug/240a/Coin%20Flip.htm
======
fractallyte
John von Neumann figured out a solution for getting fair results from a biased
coin:

1\. Toss the coin twice. 2\. If the results match, start over, forgetting both
results. 3\. If the results differ, use the first result, forgetting the
second.

This has appeared on HN before, but no one's pointed it out so far in the
discussion. More info:
[https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...](https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin)

~~~
samfriedman
Is there an extension to biased objects of more sides, say a die?

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Steko
I think if you do 6 rolls and take the sum modulo 6 it works.

edit: no, got confused with something else (game I used to play was subject to
widespread paranoia about the dice including the RNG source and whether other
players had hacks. The dev ended up jumping thru all sorts of hoops to satisfy
people -- taking random.org, letting each client add their own modifier and
then using VN)

I think the straightforward way to extend VN is rolling 6 dice and reroll all
if there are any duplicates, then use the first one (or whatever). Probably
more efficient to turn the die into a coin and then use a few coins to
approximate the die.

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kobeya
That would depend on get probability distribution I think.

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rahimnathwani
"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."

This doesn't make sense, because they started counting the number of heads
from when the coin is launched. But the first state (actually the first few
states) are irrelevant, because you always wait a second or two before
catching the coin. If you follow this line of reasoning, then the state after
the customary delay is the one that has the higher probability. But that state
is fair (50/50) and, because the customary period varies, not determined by
the initial state.

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wakamoleguy
How do you reckon that the state after the waiting period ends is fair, if, as
the article states, any point in time is more likely to have the initial side
up? It would follow that at the end of the waiting period, the initial side is
more likely to be up. And going forward from there, that new initial state
(measured from the end of the waiting period) will continue affecting the rest
of the flip. This bias will diminish over time, which backs up Premise 7 that
more revolutions reduces the bias.

On a tangential note, this behavior reminds me of Benford's law, which is
probably based on other statistical reasoning.

[https://en.m.wikipedia.org/wiki/Benford%27s_law](https://en.m.wikipedia.org/wiki/Benford%27s_law)

~~~
Adverblessly
I don't really follow the logic presented in the article.

For convenience, let's assume that the coin spins at 2 seconds per revolution.
The coin begins in a Heads up position. After 0.5 a second, the coin is on its
edge. The coin then takes another second to pass through Tails and again be on
its edge. The coin then takes another 0.5 a second to reach its initial state
of Heads up.

So it sounds to me like the coin spent 0.5 a second in "Heads", then 1 second
in "Tails", then 0.5 a second in "Heads" before starting all over again.

If I borrow the numbering analogy from the article, it sounds like if we
examine the coin in 0.5 second increments we get H T T H H T T H H T T H H,
etc. which invalidates the article's claim.

Furthermore, the article claims a 51-49 split between sides. Even if we accept
its reasoning (i.e. H T H T instead of H T T H H T T H) To get that specific
result, (if my math is right) the coin would have to flip around 25 times on
average (i.e H T H T H T H T H T H T H T H T H T H T H T H T 0.5*H), which is
a specific number not mentioned in the linked article.

All of that to say that I have no intention of refuting the paper behind the
article, nor do I think they are wrong, but just that the logic presented by
the article (and not the paper which I didn't read!) sounds too simplistic to
be correct to me.

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gowld
More simply:

Imagine a counter ticking 1,2,3,4...

If you call stop at a random time, uniform over the whole tick time, there's a
bias toward odds.

But if you choose to call stop at moment 20.5 (and stopping at a fractional
time forces the system to round to the nearest whole number), plus random
variation due to real-world physics, evens and odds are equally likely.

~~~
vorathing
I think what the parent post meant is that this reasoning is flawed by
assuming the coin spends the same time face-up in the very first iteration as
in every other one, and that's a fallacy. To put it simply, assuming a 4
second rotation, the first second will be spent in state A, the next second
will be spent equalising the time in both states, the next second will
actually put state B second ahead in face-up totals, then reequalizing, ad
vitam eternum - which invalidates the article's point about always having one
with time at most equal, and one with time at least equal, since they
alternate in total face-up time.

Edit : Basically, unless you start the coin in the vertical state, the odds-
and-even comparison doesn't apply.

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unfamiliar
I thought this was going to go in another direction: if someone proposes to
flip a coin, then they have probably decided that a 50/50 chance is more
favourable for them than any other potential decision method (like, say,
discussing what the fairest or most reasonable decision to make would be).

Hence if you agree to a coin flip that someone proposes, you have already
lowered your chances of a favourable outcome for yourself.

~~~
jimrandomh
No, because other decision methods have costs (eg time spent discussing),
separate from the favorability of the outcome. Most coin flips are done in
low-stakes contexts, where people would rather let the other person decide
than waste five minutes.

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LoSboccacc
I like the idea of the first kick in the world cup being decided by debate
tho.

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microtherion
Are you a supporter of the Greek team by any chane?

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joosters
_1% may not sound like a lot, but it 's more than the typical casino edge in a
game of blackjack or slots._

That's an exceedingly optimistic view of casinos. Slot machines have a far
higher edge than 1%, generally in the range of 3-15%.

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3pt14159
Per pull? Really? I always thought casinos were dumb, but how could they be
_that_ dumb?!

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aidenn0
Note that Slots have the biggest swing for the house.

A slot machine is basically a money-making skinner box.

Blackjack with basic strategy (no card counting, just some 0 memory rules that
would fit on a 3x5 card) is a narrow edge for the house, as is craps.

~~~
joosters
Casinos have plenty enough slot machines to average out any swings in
profit/loss. I doubt the variance is that large overall.

Some time ago, I remember reading a talk given by a Las Vegas casino operator
during a conference on risk. They analyzed their top ten largest losses for
the previous year, and found _none_ of them were associated with their
gambling results. The events that actually cost them were completely unrelated
occurrences, like their resident entertainer getting savaged by a tiger, large
fines from the gambling commission because of an incompetent employee, and so
on. The gambling side of the business generated remarkably smooth results.

~~~
thefalcon
I think in this case "biggest swing for the house" should be interpreted as
"biggest edge in favor of the house," not anything to do with variance.
Everyone knows the house always wins in the long run (that's why "the house"
is so much nicer looking [well, bigger, anyway] than your house).

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strong_silent_t
If I'm understanding the article correctly, as long as the guesser can't see
the coin, and chooses randomly, it is a fair 50/50\. I don't think you even
need to flip the coin. The coin could just be sitting on a table in the next
room. If the guesser is ignorant and chooses randomly I don't think you even
need the coin. You could just write heads or tails on a piece of paper, cover
it, and let them guess.

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iambateman
Aside from the inherent probability in the coin, it is also possible for the
coin flipper to exude bias if the coin is caught.

I can time a coin catch to get 75-25 heads.

Don’t let someone catch the coin if they flip it.

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Kroniker
How? Do you see it as it falls? or do you feel it when you catch and then
choose to do the "Flip on to your arm" thing or not?

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mikekchar
Not the OP but I can do it too. It's all timing for me. When you flip the coin
it spins at a determined rate. If you practice, you can time catching it. It's
surprisingly not so difficult.

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etskinner
Something about the evens/odds heads/tails comparison the author makes doesn't
sit right with me. If you're confident that the coin will make more than one
flip, then how much cumulative time it has spent in 'heads' position doesn't
matter (gambler's fallacy), since the 'catch' timing is effectively random
after that first flip.

~~~
seanalltogether
I think I figured out whats bothering me about it. We need to figure out at
what point in the rotation of the coin it can be determined to have changed to
the next face. Is it when the coin is perfectly vertical? Maybe even a few
degrees before because the rotational velocity will likely carry it over to
the next face after making contact with your hand?

It seems that in any case the coin does spend more time statistically on the
face that it was initially flipped from, but it's not exactly
(num_rotations/2)+1

~~~
wakamoleguy
The actual paper linked from the article does a great job explaining this.
They mention that the side facing up at any moment is a reasonable
approximation for how it would land. The time each side faces up is
significantly affected by the angle at which you flip, as the coin precesses
during the flip. Page 4 has an excellent pair of graphs illustrating this.

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yosito
When I flip a coin, I reach up and catch it in mid air, then slam it on to the
back of my other hand. I feel fairly certain that those two actions would
negate any bias discussed in this article, and quite possibly introduce their
own. If you know enough about a particular way of flipping a coin, you could
know whether the chances are off from 50/50, but if you know that much, you
could either devise a way for the result to be in your favor most of the time
or you could ask for the method to be changed to make sure the other person
isn't cheating. In either case, coin tosses are often done in casual
circumstances where close enough is good enough.

~~~
aidenn0
Never, ever, allow someone you don't trust to do this. In a day you can learn
to get about a 70%+ swing in your favor by timing the throw and catch, and if
they call it in the air, you can either flip it twice or just once when you
slam on your hand.

All fair coin tosses should land on a flat surface. Felt or carpet is better
than tile because it reduces the odds of the coin spinning when it lands,
which is weight-biased.

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skizm
I am not even kidding when I say Bill Belichick is probably reading this
article and will now have his captains try and game coin tosses.

~~~
harryh
I generally don't pay too much attention to the coin toss, but Slater
definitely called heads when the Super Bowl went into overtime.

Maybe he already knows.

EDIT: [https://www.patspulpit.com/2017/2/7/14522972/patriots-
captai...](https://www.patspulpit.com/2017/2/7/14522972/patriots-captain-
matthew-slater-has-called-the-same-side-for-every-coin-toss-for-6-years)

It sounds like it's been Slater making the all on his own but you never
know....

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fastaguy88
For an alternative view, check out Andrew Gelman's: "You Can Load a Die, But
You Can’t Bias a Coin"

[http://www.stat.columbia.edu/~gelman/research/published/dice...](http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf)

~~~
gowld
Gelman doesn't contradict. Gelman says that a properly flipped coin is always
approximately 50% heads, regardless of coin design, but NOT exact, and NOT
regardless of flipping style.

The OP is about empirically observed real-world tosses.

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lordnacho
Interesting, but I'm not sure I buy the argument. Which seems to be that
HTHTHTHTH... will always have either more Hs or equal H and T in it.

If the thing is tossed but stays in its initial state, is that a coin toss?
What if you didn't hold it either H or T up? It's not impossible to hold it
sideways and throw it up.

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aidenn0
It seems that for all biases except the weighting, all that is needed is for
the caller and thrower to be opponents, and call it in the air. Then the
thrower will want the result to be as unpredictable as possible since they
cannot predict what the caller will call.

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GuB-42
Use rock-paper-scissors instead.

It can be considered a game of skill but any one player can turn it into a
game of pure chance if he wants to, as such, is can be considered fairer.

The reason why RPS can become a came of chance is that there is no way to gain
advantage over a player who plays randomly with equal probability : the
outcome will be 50/50 no matter what. In game theory, it's called a Nash
equilibrium.

Of course, if the opponent is playing sub-optimally (i.e. not randomly), you
can try to take advantage of it.

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tzs
I recall reading once about a RPS playing robot that would win most of the
time against humans by taking advantage of its much faster reaction time and
much quicker ability to process input. It delayed making its choice until it
could see what the human was doing. The robot was fast enough that the delay
was imperceptible to the human.

Humans vary in reaction time and processing speed. I wonder if there are any
humans faster enough than average to be able to cheat that way?

Professional baseball players might be good candidates to try to train to
cheat at RPS. They have to be among the fastest humans at processing input and
reacting in order to hit major league fastballs.

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baddox
Isn’t hitting a baseball more about prediction and less about reflexes? I
would think other sports would require quicker reflexes, like boxing, fencing,
and table tennis.

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aidenn0
They read the arm-slot, grip, and initial spin of ball from 90 feet away, and
then react to it.

Table tennis is similar because you read the spin based off of the position of
their paddle hitting the ball (table tennis balls are typically unmarked so
reading the spin directly is a no-go).

OTOH, the table tennis stroke is more similar to a baseball bunt than a
baseball swing as the paddle spends more time in a position overlapping the
predicted path of the ball. This is somewhat negated by the much shorter
distances in table tennis. It's hard to say which requires more quick reaction
to complicated inputs.

~~~
baddox
Yes, they have some very small amount of time to predict the future path of
the ball based on the motion and initial behavior of the ball, and begin a
swing based on that. Apparently a fastball takes just under half a second from
release to plate, and a swing takes 150ms to execute. That gives the batter
about 300ms to decide whether to start a swing, which I'm guessing is around
the same amount of time between hits in table tennis. Of course, I don't know
how much a batter can adjust and aim the bat mid-swing. Also, some table
tennis shots give you way less time than 0.3 seconds, but those are usually
slams that would only be successfully returned if the player made a lucky
guess.

~~~
sib
For a surprisingly-fascinating look at issues related to the role of the
fastball in baseball (touching on history, physics, biology, measurement
error, etc.), I suggest watching the movie, Fastball...

[https://www.vudu.com/movies/#!content/743314/Fastball](https://www.vudu.com/movies/#!content/743314/Fastball)

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flarg
As an aside, I practiced false flipping a whole ago and got quite good at it,
assuming I remember the initial state I can always know the end state

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EGreg
Can you explain false flipping physically?

On a related note, it's much harder to fake randomness over time:

[https://faculty.math.illinois.edu/~hildebr/fakerandomness/](https://faculty.math.illinois.edu/~hildebr/fakerandomness/)

~~~
flarg
Make the coin wobble instead of rotate when you flip it, looks like rotation
but it isn't. Shiny coins work best.

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SubiculumCode
With normal people, any bias in the coin is cancelled out by lack of knowledge
of which bias the coin has.

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nixpulvis
What the original paper really needs is an extension to the world of the
spooky and quantum. Right!?

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Robotbeat
The easiest way to cheat is to just feel which side of the coin is which right
after you catch it.

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jjnoakes
And then go back in time and call it in the air?

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aidenn0
I said this in another comment, but if you catch it and flip it down onto the
back of your other hand, you can often surreptitiously add an extra flip in
without being noticed.

~~~
Robotbeat
Exactly. I've done this multiple times (always letting people in on it
afterward, of course). I have almost no skill in slight of hand, but it fools
people every time.

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dwaltrip
It seems a coin spin (on a table top) would avoid these issues.

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SamBam
The article addresses that: it's much worse.

~~~
dwaltrip
Interesting... thanks for the correction. Apparently the head side being
slightly heavier do to the extra decoration and whatnot severely biases the
coin to falling tails up when spun on its edge, up to 80% of the time with
some coins.

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JepZ
Actually, my advice for a good strategy would be to train coin flipping until
you are as reliable as the coin flipping device ;-)

> [...] an automated "coin-flipper" device capable of flipping a coin and
> producing Heads 100% of the time.

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bio_end_io_t
What type of coin are we talking about?

A coin with rounded edges won't land on its edge 1 out of 6000 tosses. A coin
with a flat edge that is thicker than either of its faces (or just thick in
general; doesn't have to be thicker than the faces) will land on its edge a
great deal more often.

Also, a coin with a heavier "tails" side will more often land on heads in a
spin.

I guess I should read the paper. Maybe it clarifies.

~~~
bio_end_io_t
The 1/6000 figure comes from a paper "Probability of a tossed coin falling on
its edge" from 1993. I looked for it in the hopes it clarified the type of
coin used, but the paper, as far as I could find, is behind a paywall.

Nevermind, found a link that talked about the paper. They used a US 5¢ coin, a
nickel which has a flat edge and I think the thickest edge out of all
(common?) US coins. At least it has the thickest edge to face ratio in terms
of width.

~~~
eric_h
I have actually seen a £1 coin land on its edge, I imagine given that it's a
fair bit thicker than any US coin, it's much more likely with a £1 coin.

