Posted: Nov 19, 2018 3:04 am
BWE wrote:Thommo wrote:BWE wrote:Thommo wrote:
Coin flips are not distributed by a bell curve (AKA Normal distribution or Gaussian distribution), they are binomial distributions.
There's an important theorem in mathematics that governs the distribution of sample means from a wide range of arbitrary distributions, known as the https://en.wikipedia.org/wiki/Central_limit_theorem, which says that for large sample sizes the distribution of sample means is approximately normal regardless of the shape of the original distributions.
I think you understood my point though.
Honestly, not really. I can't discern any difference between the first question, for example, and the question "Why does counting work?" with the assertion "There is no reason it should.".
It's very difficult to tell what the question is after, and the statement makes no sense to me at all, I can't see what it's founded on.
It's more than why does counting work but it may have the same root issue. With the coin flip the question is why does the average hover around 50%? Maybe it's feynman's why do magnets work issue but I don't think so.
Symmetry.
The universe is indifferent to objects that have indifferent properties.
If you make a "coin" that instead of being a flat disc with two identical faces is instead a hemisphere you will find it does not land heads up 50% of the time. Or if you just take a standard coin and bend it a few degrees you will find it does not land heads up 50% of the time.
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To answer in more depth:
It makes little sense to ask why coins land heads up 50% of the time and relate that to probability theory for a couple of reasons:
- The mechanics of a coin flip are well known, and essentially deterministic. Given a fixed initial upward force (at specificed g, air pressure and assuming no wind speed and that it lands on a fairly regular surface, all of which are realistic assumptions for some conditions) and initial upwards rotational speed it is possible to deterministically predict how many half-rotations take place before the coin rests, and thus which side is face up when it rests.
- The shape of an approximately fair coin is such that it has two large stable bases, which are labelled "heads" and "tails". The properties of these stable bases are roughly symmetric, meaning that if the coin contacts the ground below a certain critical downward speed and with below a certain critical rotational speed (that both vary with the angle of the coin at the point of impact) it does not have enough energy to escape the stable state of resting with the side that is currently closest to down, down.
- Those coin flips which are conducted with an arbitrary and unknown side face up have already undergone a pseudo-randomisation. A complex set of unpredictable factors will lead to you taking the coin from your pocket with a specific side up.
- Above a certain rotational speed and initial upward velocity the system becomes highly sensitive to initial conditions.
- Those shapes which do not have these properties, typically because they have an asymmetry of density distribution or shape do not have the property of landing heads up 50% of the time. Those times the flip itself is of controlled fashion (e.g. people who have practised throwing the coin upward with close to zero rotational speed) do not have the property of landing heads up 50% of the time.
- Probability theory does not only apply to events with p = 0.5. Events with different probability are covered by different distributions.
Examples:
A shape that has physical properties (and uniform density), such that it always lands with a particular side up, a long way from 50% heads and 50% tails!:
An approximately fair and symmetric object subjected to approximately fixed initial conditions will land with a large bias of one side being face down:
https://en.wikipedia.org/wiki/Buttered_toast_phenomenon
In the past, this has often been considered just a pessimistic belief. A study by the BBC's television series Q.E.D. found that when toast is thrown in the air, it lands butter-side down just one-half of the time (as would be predicted by chance).[4] However, several scientific studies have found that when toast is dropped from a table (as opposed to being thrown in the air), it does fall butter-side down.[5][6][7] A study by Robert A J Matthews won the Ig Nobel Prize in 1996.[8][9]
http://iopscience.iop.org/article/10.10 ... 4/005/meta
The orthodox view, in contrast, is that the phenomenon is essentially random, with a 50/50 split of possible outcomes. We show that toast does indeed have an inherent tendency to land butter-side down for a wide range of conditions. Furthermore, we show that this outcome is ultimately ascribable to the values of the fundamental constants.
I'm just going to round out the post (which has ended up far longer than intended), with the single word that sums the whole thing up, because it's incredibly important physically and mathematically. So much so that I can't really wrap my head around why anyone would question why laws of physics could act differently on identical states. I almost posted my reply after writing that single word.
Symmetry.