The Coin Flip: A Fundamentally Unfair Proposition? (2009) | Hacker News Reader

The Coin Flip: A Fundamentally Unfair Proposition? (2009)

econ.ucsb.edu

128 pointsscandox108 comments

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_...

If anybody's curious about the math, it's pretty simple.

Assume p is the probability of flipping heads. Then, pp is the probability of flipping 2 heads, (1-p)(1-p) is the probability of flipping 2 tails, (1-p)p is the probability of flipping tails then heads, and p(1-p) is the probability of flipping heads then tails.

So, basically p^2 + (1-p)^2 + p(1-p) + (1-p)p = 1.

Then, we ignore the first 2 terms (since that's when the results match and we start over), and we're only left with the p(1-p) case and the (1-p)p case. These are equally likely, and the first coin is heads in the first case and tails in the second.

you know what always screwed me up in stats? conditional probability. your expansion makes perfect sense (you've constructed a new measure) but I'm left here wondering if just dropping the first two terms is really the right way to find the conditional density (conditioned on the first two flips coming out the same).

Wow. I took this to the nth degree to convince a colleague. Very little variance between biased and unbiased figures:

    [javascript]
    var sim = () =>{
        var percentChanceOfHeads = 99;
        var getRandomArbitrary = (min, max) => {
            return Math.random() * (max - min) + min;
        };

        let heads = 0, tails = 0;

        for (let i = 0; i < 100000; i++){
            var test1 = getRandomArbitrary(0,100);
            var test2 = getRandomArbitrary(0,100);

            if (test1 > percentChanceOfHeads && test2 > percentChanceOfHeads) continue;
            if (test1 <= percentChanceOfHeads && test2 <= percentChanceOfHeads) continue;

            if (test1 <= percentChanceOfHeads) heads++;
            else tails++;
        }

        let total = heads + tails;
        
        let heads2 = 0, tails2 = 0;

        for (let i = 0; i < total; i++){
            var test1 = getRandomArbitrary(0,100);

            if (test1 <= 50) heads2++;
            else tails2++;
        }

        var distanceClosed = Math.abs(heads-tails) - Math.abs(heads2-tails2);

        return distanceClosed;
    };

    for(let i=0;i<10000;i++)temp.push(sim()); console.log('avg variance:' + temp.reduce((a,c)=>a+c)/10000);

Shorter version:

    chanceOfHead=0.9;
    heads=0;
    tails=0;
    flips=0;
    while(flips<1000)
    {
     coin1=Math.random()<chanceOfHead;
     coin2=Math.random()<chanceOfHead;
     if(coin1==coin2) continue;
     heads+=coin1;
     tails+=coin2;
     flips++;
    };
    console.log(heads,tails);

Sure, that will get the basics. We wanted to compare it to what an unbiased set of flips of the same number would look like over time, so there's a little more to it.

It's also interesting (and I guess should be obvious) to note that the "thrown out" tosses seem to be a pretty consistent ratio based on the "biased-ness" of the coin.

Oh JavaScript. Will a Boolean always on all browsers coerce itself into a 0 or 1 if treated like a number?

The Javascript coercion rules, arcane as they are, are actually part of the spec and (barring any bugs) should work across browsers.

That is amazing! super biased coin and it works! var h = 0; var t = 0; for(i = 0; i < 1000; i++) { var a = Math.random() < .1; var b = Math.random() < .1; if(a !== b) { if(a){ h++} else {t++}} }; console.log(h,t);

You could also just note that the two possible results where both coins differ are equally likely.

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

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.

That would depend on get probability distribution I think.

Just divide the sides into two groups. For example:

(1, 2, 5, 6) -> heads

(3, 4) -> tails

And use Neumann's algo.

You could wait until you got a run of all possible sides, but that could take a while.

I assume it's impossible to achieve in bounded time?

With a few dozen flips you can achieve a greater likelihood that you're flipping the most biased coin on the planet, and stop the game.

Right, this was a theoretical question..

The coin can only generate a limited amount of entropy per flip, it could take arbitrary long to get enough to generate 1 unbiased coin-flip.

You could bound the time if you knew the exact distribution though. Not sure if there's a method that's guaranteed to meet this bound.

How could you bound the time if you knew the exact distribution? Like say it has heads probability p?

Yeah I may have spoken to soon. There's a minimum bound, but now that I think of it there's generally no reason to believe that any subset of outcomes for any bounded number of flips would have a probability of exactly 1/2. In fact this is impossible if p is transcendental.

The set of p for which there is a method to do it in finitely many flips is quite interesting though. It is bigger than the irrational numbers (it contains sqrt(1/2)), but smaller than all algebraic numbers. At the very least you'd need 1/2 ∈ Z[p], but that might not be enough.

I don’t think this would be possible even with a fair coin.

Very cool thanks for sharing. One of those things that is non obvious until you read it, then totally obvious.

"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.

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

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.

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.

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.

Ah, you're right. This is the case if the coin is flipped with no precession. The paper linked takes precession into account based on the angle you flip the coin at, and this significantly changes the time H or T is up. The paper has two excellent graphs depicting this on Page 4.

The paper linked was also based on 27 observable coin flips, all made with the same coin-flipping machine. Out of those 27, 18 were between .500 and .504 in probabilities - and the paper concludes by saying that the "the classical assumptions of independence with probability 1/2 are pretty solid".

Edit: Actually, 21 out of the 27 flips were in the 0.5 - .505 range.

I disagree. "Any point in time is more likely to have the initial side up" if you count the initial state, which for the purpose of a coin toss you wouldn't. After a coin is flipped it's going to spend 50% of its time heads up and 50% of its time tails up.

I'm pretty sure the article is wrong. Hold on while I write a little script to test this out.

Edit: Script below; feel free to critique or make modifications where you think they are more realistic.

// Let 1 represent heads and 0 represent tails. coin_state = 1;

number_of_heads = 0;

number_of_tails = 0;

for (i = 0; i < 1000000; i++)

{

// How fast the coin flips (flips per second)

flip_speed = (Math.random() * 10) + 20;

// How long the coin is in the air (seconds)

catch_time = Math.random() + 2.5;

number_of_flips = Math.floor(flip_speed * catch_time);

/* Odd number of flips means coin ends up in opposite state from start (tails in this case). Even number of flips means coin ends up in same state as start (heads in this case) */

if (number_of_flips % 2) { coin_state = 0; }

if (number_of_flips % 2 == 0) { number_of_heads++; }

else { number_of_tails++; }

}

console.log("Heads:", number_of_heads);

console.log("Tails:", number_of_tails);

The article is wrong in that it assumes the initial state of the coin has as much duration as the subsequent states. If that was the case, the difference in duration would be the same as in the odd-even sequence :

  number       1 2 3 4 5 6 7 8 9
  odd          1 1 2 2 3 3 4 4 5 
  even         0 1 1 2 2 3 3 4 4
  difference   1 0 1 0 1 0 1 0 1

However, what happens here is different - you start at "half a state", meaning that the graph is a see-saw, as such:

  rotations    0.0  0.25 0.50  0.75 1.00 1.25 1.50
  head         0.0  0.25 0.25  0.25  0.5 0.75 0.75
  tail         0.0  0.0  0.25   0.5  0.5  0.5 0.75
  difference     0  .25     0  -.25    0  .25    0

You're right based on the article. It turns out the paper takes precession into account, which, depending on the angle you flip it at, significantly changes the time that the initial face is up.

You're right. My comment doesn't make sense.

Agreed, I think this is an overly-simplified explanation. Skimming the paper it looks like the crucial thing which introduces bias is precession of the coin.

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.

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.

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

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

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%.

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

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.

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.

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).

Nevada publishes the average edge for slot machines, e.g. http://gaming.nv.gov/modules/showdocument.aspx?documentid=76...

(Look on page six, where it breaks down the 'win percent', or the house edge, by the stake size)

Logically correct; as a game of blackjack following basic strategy is less than 1% ;-)

sure but that pays for salaries, building, drinks...etc. So they're lucky if they end up with 1%

The comparison was made to casino games, not actually running a profitable casino.

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.

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.

Or learn to feel the coin:

https://bar-tricks.wonderhowto.com/news/win-coin-toss-every-...

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?

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.

For me the trick is to know which side it was on when I threw it, and doing the exact same throw each time. You have to practice the same move again and again. Throw and catch throw and catch, like practicing music scales.

If you're doing the same throw and the same catch each time, it just becomes muscle memory (again, like playing an instrument) and you can get a good success rate.

If you: -start with the same face pointing up - apply a fairly consistent force with your thumb, and - wait a fairly consistent time before catching

Minimise speed of coin spinning and get good at consistent timing of catching.

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.

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

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.

Imagine that the differences were magnified, say heads 90% of the time. Would the randomness of the catch erase that difference?

Think about it as randomly picking (catching) from a bag that is 49% heads and 51% tails (time spent in that position in air).

And additionally...for the purpose used, it is "random enough".

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.

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.

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.

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...

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

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...

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.

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.

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.

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.

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.

Better is this implementation at the New York Times that let's you play the computer. It's amazing how it really does beat you:

http://www.nytimes.com/interactive/science/rock-paper-scisso...

Here it is: https://www.theguardian.com/technology/video/2013/nov/06/jap... or http://www.k2.t.u-tokyo.ac.jp/fusion/Janken/index-e.html

It's from the Ishikawa Watanabe Laboratory [0] at the University of Tokyo. They have a lot of cool research with high speed (1 kHz) video systems.

0. http://www.k2.t.u-tokyo.ac.jp/

It's too late to react, a human has to look for clues before hand to try and predict the outcome (which is quite possible, I've for instance noticed that a person throwing a rock will often squeeze the hand noticably harder). But changing after/during the fact will be noticable.

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.

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.

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.

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

RPS championships have been held...

https://www.youtube.com/watch?v=_eanWnL3FtM

How can a human participant play randomly with equal probability without using a three-sided die or other such source of randomness?

You can memorize a few random numbers beforehand. It is perfectly fine as long as they are used only once.

You can look for external sources in order to increase your entropy, like the seconds hand of a clock.

You can get additional numbers or limit your bias by mentally running a PRNG. A lagged Fibonacci generator is a good candidate for this.

Otherwise, you are right, you simply cannot mentally generate good enough random numbers.

It depends on how often you play RPS. If it is something that happens at most a few times a day, you could generate a random list of choices in the morning and step through it as needed during the day.

If you are on a Unix or Unix-like system, you could make a list with something like this (line wrapped for presentation):

  xxd -p -l 5 /dev/urandom |
  tr [a-f] [A-F] |
  sed -e 's/./c&dlfx0=rd1-lfx0=pd2-lfx0=s/g' |
  dc -e '16i[d3-d3-d3-d3-****]sf[[rock]psd]sr[[paper]psd]sp[[scissors]psd]ss' -

which gives something like:

  rock
  scissors
  rock
  scissors
  paper
  scissors
  scissors
  scissors
  paper

To memorize your list for the day, probably some kind of system where you have several images associated with each of rock, paper, scissors, and then use some standard memory technique, like a mind palace, to remember an image sequence for your particular list.

If you don't want to be memorizing something new each day, it would probably work to just memorize one reasonably long sequence and just cycle through it. Assuming we are talking about casual RPS here, not people who compete in RPS tournaments, your opponents probably are not going to notice that you have a pattern.

For instance, if you happen to have memorized pi to more than the average person, you could step through the digits of pi, taking each digit mod 3 to get your choice (skip the digit 0). I don't know pi very far, just 3.141592653589793238462643383, but that would give me 7 days of 4 plays per day before repeating. For things like RPS to decide who drives to lunch at work and things like that, that would probably be good enough to not have anyone notice the pattern (well...unless someone else at work was also doing pi...).

You probably could get close enough by mentally stimulating a really simple prng

You could memorize a random sequence.

The opponent almost always plays sub-optimially, lots of research on this as well for example: https://www.washingtonpost.com/news/arts-and-entertainment/w...

Need a random number generator or other computerized solution to create 'fair' 50/50 odds

RPS is definitely gamed - almost everyone that suggests RPS chooses paper first - the person presented with the option of playing the game will almost always choose rock first. Now I simply decline RPS because when I play my scissors, they end up saying 2 out of 3 man. crazy

It’s always best of three ;)

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

I would not consider that little game to be even or fair...I've learned a few tricks such that, best out of 3, I can nearly always win.

Care to share?

People will copy a winning opponent, and will not repeat themselves (usually).

such as?

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

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/

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

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

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

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

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

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.

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.

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

The article addresses that: it's much worse.

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.

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.

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.

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.

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.

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