Coin-tosses aren't fair
Many statistics examples start with "assuming a fair coin-toss..." But it turns out that coin-tosses aren't fair; depending on your toss, there's a small-to-alarming bias in the result.
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 Coin Flip: A Fundamentally Unfair Proposition? (via Schneier)2. If the coin is spun, rather than tossed, it can have a much-larger-than-50% chance of ending with the heavier side down. Spun coins can exhibit "huge bias" (some spun coins will fall tails-up 80% of the time).
3. If the coin is tossed and allowed to clatter to the floor, this probably adds randomness.
4. If the coin is tossed and allowed to clatter to the floor where it spins, as will sometimes happen, the above spinning bias probably comes into play...
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So Gildenstern was wrong?
Wouldn't the fair thing for the flip/catch thing be to just not let the person calling the coin see what side is up?
Call the coin flip, pull a coin out of your pocket without looking, flip, catch, show side, then show other side so they know it's not a fake coin. (i'm not sure how many random wonks walk around with fake coins these days)
It's interesting to see this paper going around the Internet again lately. The comments are kind of funny, too -- people raising "obvious" explanations for skewed coin tosses (like sleight of hand), thoughts that both hit and miss the point.
Persi Diaconis was a professional magician before becoming a mathematician. He's been working with the mathematics of randomness his whole career and has previously attracted mainstream attention for published work on randomness with relation to card shuffling.
Because it's easy to understand and relate to Dr. Diaconis' approach to this subject, this seems like a fantastic opportunity for people to take their understanding further. The actual papers are easily accessible here, something that isn't always the case with mainstream discussion of deeper issues in science.
I was buying the argument up until point 4, when the author lost all credibility. If it clatters, according to point 3, how could subsequent spinning make it less random?
The bias arises from the designs stamped on the coin. Even if you're using a pre-1964 silver quarter, you have the ratio of blank space to design to deal with, which becomes more of a factor the higher the relief of the coin. I would think quarters come up heads more often because George Washington's bust was rendered in much higher relief than the eagle on the tails side, which would mean more "missing" material on the heads side making it lighter.
This is why the spots on approved dice at Vegas are painted on rather than drilled into the surface. I would think the dice in your Monopoly set would be inherently loaded to roll sixes because dice are designed so that opposite sides always add to seven, which would put the heavier "one" side opposite the lighter "six."
Of course, I could be completely full of it. Anybody else?
If you spin the coin leaning slightly on the edge of its rim, it will always land on the side corresponding to the opposite edge of the rim (provided it doesn't hit anything and carries on spinning on a flat surface until it comes to rest)
The thicker the coin the easier this is.
Tails never fails!
try this:
Note the face up as you rest the coin on your curled index finger and thumb. Flip it.
Now catch the coin as it falls in front of you with your flipping hand now out at elbow height, palm up. Snatch it and smack it over onto the back of your other wrist. Note the face up result.
Alternately, catch the coin in a palm-up, overhead snatch and smack it down onto the back of your other wrist. Note the result.
Surprisingly consistent.
Rock Paper Scissors (Lizard Spock) ensures the craftier person wins, which is more than fair. Also, (almost) everyone carries the proper equipment around with them at all times.
Guildenstern: "Consider: One, probability is a facter which operates *within* natural forces. Two, probability is *not* operating as a factor. Three, we are now held within un-, sub- or super-natural forces. Discuss."
Rosencrantz: "What?"
Hey, life isin't fair...why should a coin toss be?
10 minutes practice of snatching the coin out of the air & giving it a crafty rub with your thumb as you slap it onto the back of your other hand will allow you to call it (or signal to your shill) right every time. It's surprising how many people you can fool with it.
I used to be part of a competitive sport and openly told people one of the ways that i was winning an advantage was to guess coin flips. No one believed me even as i seemed to win every important coin toss and usually kicked their butts.
Sometimes people want to believe so hard they refuse to see what is happening before their eyes.
I prefer the highest percentage technique of all:
headsiwintailsulose! (quickly flip coin)
so how's this work in Two-Up?
(worth it for the jargon alone)
http://en.wikipedia.org/wiki/Two-up
Note that that estimate of the 1% bias towards the face it was thrown on has not been empirically tested, and according to the paper, would require 250,000 coin tosses to test with any kind of statistical significance:
"Our estimate of the bias for flipped coins is p =˙ .51. To estimate p near 1/2 with standard error 1/1000 requires 1/root(2)n = 1/1000 or n.= 250, 000 trials. While not beyond practical reach, especially if a national coin-toss was arranged, this makes it less surprising that the present research has not been empirically tested "
In other words, tossing and catching may have a tiny bias, but even taking that into account is random enough for most purposes. (And if you want to grift someone, spin the thing.)
A (tangentially) related point is that bias in any coin can be corrected by a simple procedure.
Toss the coin twice, until as your result you get either HT or TH. Call the former tails and the latter heads, and the sequence of results will be absolutely fair.
The tradeoff is typical for algorithms of this sort, in that the worse the bias, the more flips are required for each result in the fair sequence. The greater the bias, the more likely we obtain HH or TT, results that force us to try again.
Even a biased coin can be used for a fair flip.
The trick is in using two flips.
One person calls "heads-tails" the other calls "tails-heads".
If the coin comes up tails-tails, or heads-heads, you retoss.
Simple statistics and probability. Even if the coin is a trick coin, and weighted for say 80-20, a tails-heads is (.80 x .20 = .16) and a heads-tails is (.20 x .80 = .16). All other combos result in a re-flip, which basically resolves back to the above two final scenarios, giving each person a 50-50 chance.
Nice "R & G are Dead" references above...well done.
Did the experiments include the effect of Harvey Dent's charred coin on spinning bias?
"The only fairness in a cruel world is chance."
In my sixth grade math class we tossed a penny a hundred times and tallied the results. Heads came up something like 58 vs. tails 42, and I have called heads ever since.
all well and good I suppose, but we all know in real life the toss is whatever the flipper says it is.
http://www.youtube.com/watch?v=wkh6if8TL2U
@ChunkyMonkeyBrain: I walk around with a fair half-dollar at all times. It's a great flipping coin, but with your protocol, if I want to win, I will win. Go read the comments on Schneier's blog to find out why.
If you want fair, you have to flip, catch, hold coin tight against the wrist, other person calls it, then show and show reverse. Make sure you watch carefully, because I know what the coin is showing when it's on my wrist...
(Of course, most people don't understand the finer points of that, so if I want to win, I'll just say "call it in the air!" and no one will be any the wiser...)
Also, the coin manipulation problem makes any of the two-toss protocols insecure, because there is no way to point where you can put the call to prevent the flipper from manipulating the coin while maintaining perfect 50/50 odds.
Before a football (soccer) match, the "coin" is standardised as a balanced object and must fall to the ground. It is also thrown by the referee to minimise cheating of the kind mentioned above.
@Anonymous:
So if you have a perfect flip but it lands and spins, it will be more likely to fall on the heavier side.
From the article:
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.
This appears to me to be completely and utterly bogus.
The only reason that there might be one more heads is that he decided to start counting on heads. Why did he start counting on heads? Because that was what was showing in the beginning, when it was resting on the thumb.
But you could equally argue that that is the very worst place to start counting. A coin flip isn't a coin flip if it doesn't leave the thumb. Indeed, it has to have at least one flip. Therefore, it is completely impossible that the first H would ever be counted.
So you could make an equally good argument saying
This argument sounds just as reasonable as the first one, if not more so, which implies to me that the entire argument is bogus.
Note that this argument is not used by the original authors anywhere in their article. Indeed, if it were the basis of their research, which it is not, then they would show a bias of much, much more than 51%, because the coin would have to flip at least 100 times before the effect of that "first side" bias, were it true, was as low as 1%.
The point being that the "memory system" doesn't exist for coin tosses: the possibility of a head is (trickster avoided) the same as a tail. If I toss 5000 heads in a row, the likely hood is that the next toss may be a tail. Prior experience has nothing to do with chance.
This is why roulette tables have signs showing what happened prior: three reds must mean a black... and so on.
@Samsam, the second argument is correct. However, the coin is still biased.
Unfortunately for the article's author, the paper (PDF) says that's not why. I mean, it's a nice mathematical way to look at it, but it's also wrong, and explicitly debunked on the paper's first page in reference to earlier work:
The problem is that assumption: It's wrong. It's a good approximation, but the following assumption that the axis is distributed normally around the perfect case is false. The axis is skewed from that assumption in a predictable and unequal manner, since you're always hitting the side of the coin away from the initial top.
Thanks, 3D dynamics engineering course! Gyroscopes are mind-bending.
superconducting electromagnets.
@Samsam again: Whoops, missed your last paragraph! My first line would have sounded a little different.
Anonymous @13:
Same here, except for the part about telling them what I was doing. You may not be able to call a tossed coin correctly every time, but with a little attention and practice, you can sure improve your incidence of winning calls.What I noticed when *matching* 50 cent pieces (remember those?) was that if flipped with a certain velocity and grabbed at a certain height and slapped to the back of the other hand, it would be the same side of the coin that was facing up at the flip, maybe 70-80% of the time. Or was it the opposite side? It was so long ago...
Anybody got a 4-bit piece?
"To provide smaller denominations, silver dollars were cut into eighths, or "bits". Thus, twenty-five cents was dubbed "two bits," or two 12.5 cent units, as it was a quarter of a dollar. Correspondingly, the terms "four bits" and "six bits" referred to fifty and seventy-five cents, respectively. For example, "Six-Bits Blues" by Langston Hughes included the following couplet: Gimme six bits' worth o'ticket/On a train that runs somewhere….
Because there was no one-bit coin, a dime (10 ¢) was sometimes called a short bit and 15c a long bit."
http://www.youtube.com/watch?v=KyqwvC5s4n8
From my experience, if you flip a coin briskly and effectively at a reasonable height and then catch it at the same height you flipped it from, 9/10 it will be the opposite of what was face up when you started.
So if you're "Heads Up" before the flip...
catch it at the same height you flipped from.
it will be "Tails Up". most of the time.
if you practice you're flipping speed. try it.
Coin tosses can be rigged also.
If nothing else, I just flipped a quarter. I attempted to keep it on the tails side as many times in a row as I could and ended at 8. My record is 18 with heads.
It all depends on how consistantly you can flip the coin at the same speed, and at what time you catch it. The sound of the coin generally works for an indicator of when to catch it.
Through that, you can also rig it so that you can catch it flipping opposite.
Once, on a certain pharmaceutical, I got 63 heads in a row, talking through each flip like a sideshow barker. Whatzit gonna be, itz gonna be heads, and...oop, heyyy! Heads it is, here we go again...heyyy! On the 64th toss, I felt "it" go, and it came up tails.
Shoulda been in Vegas that weekend, I guess, but it's probably good that I wasn't.