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Science of the coin-toss: Bias in Heads-or-Tails
MrSharkey writes " An interesting
article published in Science
News puts a new scientific spin on the outcome of the venerable
coin-toss. "A new mathematical
analysis suggests that coin tossing is inherently
biased: A coin is more likely to land on the same face it started out
on.""
Perhaps related, bread more often falls butter-side down because it usually only has time to complete half a rotation in the distance it falls from your countertop.
A coin is more likely to land on the same face it started out on.
If this is true, we would still want to call the opposite face since we after it lands, we always flip it onto the other hand. That is, if we start with heads facing up, and it lands more frequently with heads facing up on our palms, by the time we slap it onto the back of our opposite hands, tails is facing up!
Yeah, guys, 51% is really biased there... especially when you can completely solve this by the simple expedient of not looking at the coin before you toss it. (or by having one person pass the coin over, and the other person call it)
pb Reply or e-mail; don't vaguely moderate.
An interesting alternative is to flip the coin so that it lands on a smooth floor, spinning on a vertical axis. Then the uneven distribution of mass between the head-side and the tail-side will cause a bias.
It is my experience that dimes and quarters are nearly unbiased for this test, whereas nickels are heavily biased (pun intended) toward tails . [In a past life, I taught a statistics class for which I assigned daily homework, deciding whether or not to take it up on the basis of a coin flip at the end of class. On days for which I really didn't want to spend all evening grading papers, I would use a nickel; I'd use a much-fairer quarter on other days. And none of the class caught on... ]
"My opinions are my own, and I've got *lots* of them!"
In football games they let the coin hit the ground and bounce around before coming to a stop. That introduces complexities that the scientific study did not address.
It may very well turn out that the odds of getting heads/tails after letting the coin fall on the ground are still 50-50.
This was on NPR the other day. There are a number of issues one of which is that when we flip a coin it actually has a fairly high probability of never actually "flipping" end-over-end. You can test this by attaching a ribbon to the coin and to, say, the table, flipping the coin and then counting the number of twists in the ribbon. It isn't a question of trying to flip it more or less times so much as the physics of the flip. As mentioned in other posts, letting the coin bounce on the ground does make things better since in that case the coin's motion is less predictable.
I'm a magician, and a "mentalist". That means, I pretend to have psychich powers (which I don't, but I don't explain that until after I've convinced the spectator that I have).
One of my tricks is to predict the outcome of a cointoss. I start out with pseudo science explanation, and then, as I continue to be correct, continue on to a supernatural explanation.
The explanation given in this article, as to why a coin is biased, can be boiled down to this (quote from the article): For a wide range of possible spins, the coin never flips at all, the team proved. . That is - the extra bias is towards the side that was up from before the toss, and is a result of the coin not spinning at all. If that's their big scoop, I'm dissapointed, because if the coin doesn't spin, it's not within my definition of a coin toss.
The article actually mentions magicians: Magicians and charlatans may take advantage of this illusion. Keller observes, "Some people can throw the coin up so that it just wobbles but looks to the observer as if it is turning over."
He has obviously seen a magician to the same trick I do. Of course I wont reveal the secret, but I can tell you this: he's wrong. The dirty work does not happen in the toss. The coin actually do spin, and the secret move is done at an offbeat moment.
Some level of added insurance would be provided by simply not allowing those selecting a landing side to see the side on which the coin begins. If the flip is being done by a third party, of course, there's the danger that there's collusion between the third party and one of the participants prior to the toss, even for a 1% better chance in the throw, but we still have a better chance of non-tampering and non-bias as a result. And regardless, even in the worst case scenario, where the participants know the side on which the flip is beginning, we only have a 1% statistical advantage to the one side. Furthermore, a non-level, somwhat randomly varied surface onto which the coin is tossed, rather than a plane, will add another randomising factor.
Most Interesting Part of the Article:
:-)
This slight bias pales when compared with that of spinning a coin on its edge. A spinning penny will land as tails about
80 percent of the time, Diaconis says, because the extra material on the head side shifts the center of mass slightly.
Is it time to start making some bets with some friends?
Wow, Taco, about 7 Slashdot readers will even get that. +1, Obscure!
That was a pretty funny book, actually.
Except that it was also a movie that more than a few people have seen. Not really that obscure.
Another case of duh. I observed this in High School in the late 1980s when my friends and I used to play various quarter games for money. It greatly increased my chances when spinning a quarter for money during lunch.
I also used it to increase my chances when playing same/different with another player. Each person spins a quarter, and both players stop their respective quarters wihtout letting the other see the results. The person can look at their own results, and one person guesses whether the quarters are they same or different. If the person guesses correctly, then they take the money. Otherwise, the other person takes the money. Other amounts of money oculd be bet, but only quarters were used to spin in the game. You can really gain a psychological advantage over a person when you win a few without looking at your results and winning each one!
At the next eco-hypocrisy-meeting, count the private jets used to get to the meeting. Should be interesting to see that
There is a neat trick for dealing with a biased coin in a coin toss:
- Flip twice.
- Discard the pair of throws if it's both heads (HH) or both tails (TT).
- Count HT as heads, and TH as tails.
(I think this idea was from John von Neumann.)
Applied to the current situation: Flip twice, once starting H down, once with T down.
I remember my middle school science teacher would have a "coin tossing" lab each year with students, students would keep track and submit the totals. It was all a lesson in probability. He had everyone use pennies dated after 1982 (when they changed the alloy). Heads up was almost 51% of the time. His theory was that heads was "rounder" than tails and that accounted for the difference. Course, 7th grade students don't exactly make the best objective testers
This is just a result of standard statistics. This has been known in Gambling for some time. If a game has a 50/50 chance, and you start losing, you are most likely to keep on losing. You are starting the next game that has a 50/50 chance of winning, however, *YOU ARE ALREADY LOSING*. The same goes on the flip side. If you are already winning, and you continue a game where you have a 50/50 chance of winning, *YOU ARE ALREADY WINNING*
Think about this. The coin first lands on tails. On the next two throws, it's 50/50 chance of tails or heads. Thus, if it landed once on tails, and once on heads, you have 2/3 tosses tails, and 1/3 toss heads.
However, statistics also says, the more you play the game, the more the overall outcome will get close to 50/50. However, if you start out losing, you are more likely to stay losing. You will just get closer and closer to 50/50 even if you don't win overall.
This is one of the number one myths of gambling. Just because you've been losing, doesn't mean your "luck" will change and you can start winning. In fact, you are more likely to stay a loser overall.
It would very useful to learn how to flip a coin (into the air), but not have it actually flip (end-over-end) as per the article. They implied that if the coin is oscillating or wobbling, people would not notice that it's not actually flipping. This could win me a lot of root beers!
The Russians have won. They have made the world a cesspool of distrust, greed, fear and hate.
There is a rather simple way of generating a "fair" event (i.e. probability 1/2) using an unfair coin. Instead of calling on a single toss, you call on a sequence of 2 tosses (H on first, T on second OR the other way around). You toss the coin twice - and reject the pair of tosses if you observe both H or both T. Even if the coin is biased - the probability of HT versus TH are equal. (This of course does not address the question of "Does the starting side have a greater probability of showing up finally?" - but that is now irrelevant. You always start with the same side showing up since the fact that the toss is biased is no longer of any consequence.
This is a very common idea in statistics - the "order" HT versus TH is what is called an "ancillary statistic".