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This is a question that I practically encountered while I was playing a game: Is tossing a coin a deterministic experiment? It might seem silly to ask but I had some thought over it.

By the definition of the term and apparent look, it is not. But I have a different notion. Deterministic experiments have the same conditions (it may be physical or regarding apparatus) while conducting an experiment. But while tossing a coin do we always apply the same force at the same point of the coin. We toss with the same side of the coin upwards.

If we do so, then then the outcome would be same. A 10 kg block would move with 10 m/s² acceleration when a force of 10 N is applied. Similarly, if we toss a coin applying the same torque at the same point, then the number of rotations in the air would be the same and give a consistent result.

I was scolded by my teacher on asking this question.

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  • $\begingroup$ Please try to answer so that a high school student can get it. If that is not possible please don't mind to post the needful explanation. $\endgroup$
    – user693540
    Mar 5 '20 at 11:36
  • $\begingroup$ Yes, it's a good question, and this is part of the reason there's been so much discussion about the meaning of probability. In practice if someone is going to toss a coin 10,000 times, the best prediction you can make is that the result will be heads about half the time. And if you want to predict how often other events occur, and you want your predictions to be self-consistent, then you need to use the rules of probability. The rules of probability can be viewed as just a system for making self-consistent predictions about how often various events occur. $\endgroup$
    – littleO
    Mar 5 '20 at 11:47
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    $\begingroup$ With an appropriately generous definition of deterministic, the answer is probably yes. Quantum effects really don't much enter into a coin toss. To be sure, the forces are sufficiently complicated to ensure that actual prediction is generally not possible. Though I have seen sleight of hand artists who can do a pretty good job of "forcing" a desired outcome on a toss. $\endgroup$
    – lulu
    Mar 5 '20 at 12:12
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    $\begingroup$ I think this question would be better suited on the Physics Stack Exchange site. Your question is effectively "are physical processes deterministic". The answer for a mathematical coin would usually be built into the setup of the problem (e.g., the assumption that it the coin is fair). $\endgroup$
    – Jam
    Mar 5 '20 at 12:41
  • $\begingroup$ See (Physics Question 63811) and (Physics Question 477910). $\endgroup$
    – Jam
    Mar 5 '20 at 12:43
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I think a way to understand this is to compare to another deterministic transformation: the baker's map. If we always choose the same initial point in the square and apply 30 times the baker's map, the image of that point in the square will always be the same, in this sense, it is perfectly deterministic. But if we have the slightest inaccuracy in the initial position, the position after 30 iterations of the map becomes unpredictable, and has all the appearance of randomness.

Systems where small variations of the initial data implie large or unpredictable variations of the final outcome are common in mathematics. One often cites the butterfly effect in chaos theory to illustrate this. Many non linear hyperbolic systems of mathematical physics have the same behavior.

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  • $\begingroup$ The relation between chaos and stochasticity is much subtler then you make it sound. In particular, the "randomness" in a coin toss stems from the fact that we don't model a very large number of DOF in the problem (either since we can't or don't want to), whereas the chaotic system still has the unpredictability aspect with a very small number of DOF (as few as one!). Also, it's possible that if you did care to fully model the coin toss experiment, you could get a good approximation even from only approximate knowledge of the initial conditions. Which is of course untrue for a chaotic system $\endgroup$ Apr 1 '20 at 9:04
  • $\begingroup$ @H.Rappeport You are correct of course and the above explanation is only a first step towards understanding the phenomenon. An interesting experiment would be to partition numerically the space of initial conditions in two regions that lead to heads and tails throwings. This could be done with some simplifications in the number of DOF. It would probably reveal that the granularity of this partition is much smaller than the attainable precision by hand throwing. $\endgroup$ Apr 2 '20 at 13:24
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This is a good question and should always ask questions like this if you want to want scientific answers! Dont get discouraged by one teacher who does not like to be challenged.

To your question: Yes you are right. If you would build a robot that always uses the same force in a vacuum with no wind and the exact same coin the result would always be the same.

HOWEVER: In normal life there are so many small things you can not control that the result seems to be random every time.

So for all purposes of statistic evaluations and computations actually tossing a coin (or rolling a die) will be a random experiment.

One interesting sidenote: Creating random things is REALLY hard to do for a computer! They do exactly the same things every time. Because of this, computers create pseudorandom values that can be exactly reproduced by the computer but seem to be random for the human eye.

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    $\begingroup$ It is tangential and straying away from mathematics but there are some factors that still make your example potentially non-deterministic: (1) the unknowable quantum mechanical positions of the particles of the robot and coin, (2) the degradation of the materials, (3) the creation of virtual particles in the vacuum that collide with the coin, etc. $\endgroup$
    – Jam
    Mar 5 '20 at 12:36
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    $\begingroup$ @Jam: The quantum mechanical position is not unknowable, it simply doesn't have a definite value before measurement (and given that, as far as we know, position is a continuous variable and therefore never completely measurable, this means the position never has a definite value). It's a very important distinction. $\endgroup$
    – celtschk
    Mar 6 '20 at 8:08
  • $\begingroup$ @celtschk Yes, that's what I meant. "Unknowable" in the sense that we can't specify position to arbitrary certainty, I didn't mean to imply it exists. I suppose something like "obscurity" or "inexactitude" might fit better $\endgroup$
    – Jam
    Mar 6 '20 at 8:37
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    $\begingroup$ @Jam: "Obscurity" again would mean that it is just a lack of knowledge. The technical term is “uncertainty”. $\endgroup$
    – celtschk
    Mar 6 '20 at 8:54
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I teach my class that tossing a coin is a deterministic experiment.

First though, you need to fully define "tossing of a coin". I also don't fully define this, but essentially I say "repeatedly toss the coin under the same conditions". In these conditions, I include:

  • same coin (no loss/gain of mass for this coin)
  • same side of the coin always starts up
  • same force applied resulting in same rotational velocity, height, etc.
  • same external conditions, e.g. no wind, same atmospheric pressure, etc.
  • and so on

Since quantum effects are negligible, the coin should end up with the same side up, i.e. it is a deterministic experiment.

Now, when most people talk about "tossing a coin" they don't stipulate that all these experimental conditions have to be the same and thus there is a random experiment. In my class, I point out that it is only random because of our inability to exactly replicate the experiment.

As a fun aside, a colleague in graduate school, who also happened to be a magician, told us that he could toss a coin with a much higher than 50% chance of getting a head. He managed to flip the coin about 10 times and got a head every time.

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This is a great question. Keep asking these questions, but don't expect your teachers to always get you ;)

Uncertainty is a statement about our ignorance of the true facts, rather than a property of the facts themselves. This is a Bayesian idea, and receives a book length treatment in Jaynes's Probability Theory: The Logic of Science.

Ponder this: I get you and your teacher together. I explain that I will flip two coins, and will want to know the probability of two heads. Before you answer I flip two coins, hiding the results with my hands. I show one to you, but not your teacher; it's a head.

A first Jaynesian analysis proceeds like so:

  1. Coin flipping is roughly symmetric (i), so H/T on a single coin toss is equally likely.
  2. There are two possible outcomes per toss (ii), giving probability 1/2 each.
  3. The coin flips are independent (iii), so HH, HT, TH, TT are equally likely giving 1/4 a priori probability to HH.
  4. Conditional on one H, we're back to a single coin toss.

So you will likely answer 1/2 and your teacher 1/4 because your probability is conditional on a different set of information.

Now (i - iii) denote points where other information could easily enter in:

i) Are the flips really symmetric? Do we have a high speed camera capable of capturing the initial conditions? Or, more likely does the flipper have an incentive to produce biased coin flips?

ii) Are there really only two possible outcomes? Could the coin land on its side? Could the experiment be disrupted? Sound like a stupid concern? Go search for censored data in survival analysis.

iii) Are they independent? If flipped by a machine loaded the same way will they be biased? Sound like a stupid concern? Go search for batch effects in GWAS.

In Jaynes' approach these questions never really stop, the only question is when you get to the point where it's not practically worth while incorporating more information.

Back to your question. How much do you know about the coin? Do you have a 3D profile of its mass distribution? Do you know how much air resistance its materials create? The air currents in the room? How precise is the flipping mechanism? How does the landing surface interact with the coin? How much time, capability and inclination do you have to actually use all that information if you had it? The less you know and the less you care, the closer you are to p=0.5. Deterministic doesn't mean you personally can be certain of the answer ;)

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  • $\begingroup$ Good answer. But "Jaynesian analysis" sounds a bit off.. Jaynes, while being one of the clearest thinkers and explainers of Bayesian analysis, is by no means the originator of these ideas. Also, "H/T on a single coin toss is equally likely", might be worth emphasizing that this statement refers to a given subjective viewpoint. Namely, one with incomplete knowledge of the initial conditions and/or the relevant physics $\endgroup$ Apr 1 '20 at 9:09
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Have you ever tried to throw a knife at a wooden board so as the knife sticks into it? I'm sure you didn't even get the knife to hit the board with its blade at start, let alone to stick in it. The angle, at which the knife hits the board, as well as the point of collision, seem pretty random initially. That's because we seem to have no control over it.

However, as we try over and over again, we learn how to feel the balance of the knife in our hand, how to adjust the force, rotation and direction, and gradually we manage to stick the knife into a board again and again. Professional knife throwers do that with astonishing certainity and unbelivable precision! Man can do that with the same knife, the same eye, the same arm and the same brain, which initially seemed to get unpredictable, random results only... One can't explain it to others – listening about force, torque, aiming and balance doesn't help in learning; but one can learn it by exercise. So we can see it is possible to control the initial conditions of a throw with some level of precision.

Be aware though, that a result depends on an initial precision and on the amplification of the initial uncertainity during an experiment. Many people can throw a coin or a die so that it lands on a chosen side, if the coin or the die makes no more than one or two rotations. However, if the object performs several rotations, or even worse, it bounces off the table several times, then the final position becomes truly unpredictable. That's because the change in the linear and angular velocity during a bounce depends greatly on the initial velocity and on the angle of collision; as a result the dispersion of states after the bounce is much bigger than before.

Simply speaking, an initial height and vertical velocity of the coin determines the time of fall, and that time times the angular velocity makes the number of rotations. Suppose you can control the coin's angular velocity precisely enough to toss a coin so that it makes one rotation, give or take 1/5 of a full turn. Then you can trick others by getting (almost) always a desired result – a fifth of the full turn doesn't change the result.

Now toss a coin with the same angular velocity, but at a height 25 times that in previous toss. The coin's fall lasts 5 times longer, so instead of $1±0.2$ rotation it will make $5±1$ rotations – and you can not reasonably predict in which quarter of that $\pm1$ range it will stop. That is, whether it lands on heads or on tails.

Whether the physical process outcome is random or predictable depends both on the precision of initial conditions and on the stability of the process itself (see the baker's map mentioned by Gribouillis). With a coin toss, just a few rotations is usually enough to transform a minor initial uncertainity into a true randomness of the result.

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