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It is now well known that a random coin toss has 1/6000 probability of landing on its edge. So the out-dated model that a coin toss always land on either heads or tails with probability 1/2 is wrong. Further, it has also been revealed that the physical coin toss process is not random, but deterministic. So the coin toss should be described as a phenomenon in Newtonian dynamical systems, and the "randomness" comes from the sensitivity of initial data, so it is not really random at all.

Therefore it clearly does not fit into the definition of classical probability. Because if we are talking about the coin toss as a random event with 1/2 for either heads or tails, then this is taking conditional probability excluding the case of landing on its edge already.

While the heads and tails had a pedagogy advantage for being easy to understand, I believe this is not an appropriate example that should appear in the textbook. And we all know there are a lot of good examples in quantum mechanics.

So my question is, why we still mistaken coin toss to be an example of classical probability? To me this is almost the same as leaving the mistake to the next generation while science has proved it is not true.

Reference:

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.48.2547#abstract

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migrated from mathoverflow.net Oct 12 '14 at 20:24

This question came from our site for professional mathematicians.

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    $\begingroup$ So there's a big difference between tossing a coin and dropping a coin (which is what the cited article is about). I have probably seen 1000 coins tossed in my lifetime, and never seen one land on its edge. If the frequency were anywhere close to that claimed in the posting, many of us would have seen coins land this way. $\endgroup$ – Anthony Quas Oct 12 '14 at 19:48
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    $\begingroup$ Are you really arguing, in a deterministic universe, that we would have no use for a theory of probability? Even if a process is deterministic, if we have incomplete data about initial conditions (most of us do not have sensitive enough thumbs to determine a coin flip), then probability is useful for modeling the situation. $\endgroup$ – Steven Gubkin Oct 12 '14 at 19:52
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    $\begingroup$ "It is now well known that a random coin toss has 1/6000 probability of landing on its edge." [...] Compare the abstract: "An experiment is reported in which an object which can rest in multiple stable configurations is dropped with randomized initial conditions from a height onto a flat surface. [...] Extrapolations based on the model suggest that the probability of an American nickel landing on edge is approximately 1 in 6000 tosses." $\endgroup$ – NAME_IN_CAPS Oct 12 '14 at 19:52
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    $\begingroup$ You say "a random coin toss has 1/6000 probability of landing on its edge" and then "the physical coin toss process is not random, but deterministic". This is self-contradictory, I suggest you first clean up what you really think about the issue before asking such an aggressive question. $\endgroup$ – Benoît Kloeckner Oct 16 '14 at 19:59
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    $\begingroup$ I'm voting to close this question as off-topic because it's not a legitimate educator question. OP is simply looking for an argument. $\endgroup$ – JoeTaxpayer Nov 15 '16 at 13:57
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When mathematics is applied to the real world, it is always only an approximation. Usual probabilistic model for the experiment of coin tossing is a good approximation. Good and useful. So there is nothing wrong with teaching this example.

Arguing like you do, we should not teach classical mechanics, because it is only an approximation to reality, and teach only quantum mechanics and (general) relativity.

This is a wrong approach. Because classical mechanics works perfectly in many practical problems. And probabilistic model works very well when applied to coin tossing, card games, etc.

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  • $\begingroup$ Well, as an aside, quantum mechanics is only an approximation and I defy anyone who says otherwise! On the other hand, the theory has a miraculous quality of revealing phenomena of nature which, to my mind, would be rather difficult to guess otherwise, and humans haven't really got to the bottom of it yet and thus it sometimes looks like it is the "truth" rather than "just a model". (You might well use classical mechanics to calculate the best trajectory of a rocket to great precision but I think you already began with an idea about how the rocket would behave, right?) $\endgroup$ – P.Windridge Sep 26 '16 at 19:07
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One often sees the phrase "a fair coin." So, if you wish, this is a theoretical object which does not exist in the real world but the name is evocative and useful. In real life a (real) coin flip is used to decide some things (football). One may object to an element of chance in a contest of skill. Perhaps the side winning the toss gets an unfair advantage. But no-one worries that the side calling the toss gets an unfair advantage. In the world of theory no-one outside a middle school science fair is tossing actual coins. Then a fair coin is a useful name for an imaginary binary object which behaves exactly like the best first order approximation behavior of real coins (or at least what we would like it to be).

What do you see as the problem?

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Stating that it is "well known" that the probability of a coin landing on its side is 1/6000 is incorrect. In fact - probability is determined by a variety of factors. In this case, the coin landing on its side would depend on the nature of the surface upon which it was landing most notably, as well as the nature of the coin's surfaces. In fact, taking all factors into account probability is actually very predictable. If you were able to model the majority of forces and masses acting on the coin as it was tossed, you would (to a limit) essentially be able to predict the outcome with a very high accuracy. Such a thing could be done with a computer. In fact, probability is more the percentage occurrence of an event within a group of large events, rather than something that can be applied to one-time occurrences. For example, last week's lottery numbers had an immensely small chance of occurring. Some would say that if the lottery was played once that the chances of those numbers occurring would make the event a miracle almost. However, since we have the lottery every week, this become normal - obviously those numbers might occur some week (it just happened to be last week). Your comments about quantum mechanics are interesting: read the last part of this blog post. http://lifephysics.blogspot.co.uk/2014/03/computers-don-always-do-what-they-told.html

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  • $\begingroup$ I've seen once in my life a coin landing (and then standing) on its edge. It probably had to do with some glitches between the wooden planks on the floor ... $\endgroup$ – kjetil b halvorsen Apr 20 '17 at 9:13
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Probabilistic models are used with great effect in many deterministic settings, and also in settings where we freely admit that the model is wrong (e.g. because events of small probability are ignored!). In fact this is the most common scenario!

It's easy to miss the point of modelling. "All models are wrong, but some are useful". Logic, math, probability are models. We study them because they're useful. The definition of "useful" here is situation-specific and open to debate! (There are other reasons to study abstract models too, e.g. for fun!)

Anyway I say your question is ill posed, because it isn't really a "mistake" to model a coin flip using a Bernoulli(1/2) random variable. On the other hand, I think it's good you're thinking about the limitations of your models and not just blindly assuming they're "right"!

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