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  1. Special Relativity/Simultaneity, time dilation and length contraction
  2. Zeno’s Paradoxes
  3. The Arkansas paradox « Statistical Modeling, Causal Inference, and Social Science
  4. of time volume Ebook
  5. The Arkansas paradox

If we want to discover the events that really happened at the same time we would need to subtract the time taken for the light to get to us. This would be highly necessary if you were observing events on the Moon: if you were on the Earth and saw the time on a lunar clock you would know that the real time on the moon was more than a second later.

But would this be enough? What about the relativistic phase differences between clocks due to motion? Special Relativity introduces yet another factor, in addition to the travel time of light, that upsets our knowledge of which events are simultaneous. The relativistic phase differences between clocks are tiny at the distance of the moon but have the startling consequence that at distances as large as our separation from nearby galaxies an observer who is driving on the earth can have a radically different set of events that are simultaneous with her "present moment" from another person who is standing on the earth.

The classic example of this effect of relativistic phase is the "Andromeda Paradox", also known as the "Rietdijk-Putnam-Penrose" argument. Penrose described the argument:. How can there still be some uncertainty as to the outcome of that decision? If to either person the decision has already been made, then surely there cannot be any uncertainty. The launching of the space fleet is an inevitability.

Notice that neither observer can actually "see" what is happening on Andromeda now. The argument is not about what can be "seen", it is purely about what different observers consider to be contained in their instantaneous present moment. The two observers observe the same, two million year old events in their telescopes but the moving observer must assume that events at the present moment on Andromeda are a day or two in advance of those in the present moment of the stationary observer.

Incidentally, the two observers see the same events in their telescopes because length contraction of the distance from Earth to Andromeda compensates exactly for the time difference on Andromeda. This "paradox" has generated considerable philosophical debate on the nature of time and free-will. The advanced text of this book provides a discussion of some of the issues surrounding this geometrical interpretation of special relativity.

A result of the relativity of simultaneity is that if the car driver launches a space rocket towards the Andromeda galaxy it might have a several days head start compared with a space rocket launched from the ground. This is because the "present moment" for the moving car driver is progressively advanced with distance compared with the present moment on the ground.

The present moment for the car driver is shown in the illustration below:. The net effect of the Andromeda paradox is that when someone is moving towards a distant point there are later events at that point than for someone who is not moving towards the distant point. There is a time gap between the events in the present moment of the two people. The "Twin Paradox" derives from an article by Langevin who used travel to a distant star and back to describe the relationships between times in different inertial reference frames.

Langevin's original example was called the "Clock Paradox" and showed that a space traveller who travels to a distant star and back finds that he has aged less than the people who stay on Earth. Benguigui provides a detailed account of how, over time, this example became the story of two twins, one who travels out into space and one who stays at home, and how it became described by Weyl in as the "Twin Paradox".

The "Twin Paradox" is an interesting example of the relativity of simultaneity and time dilation and deserves close study so that these features of Special Relativity can be understood. However, be warned, the reason that the "twin paradox" has attracted so much puzzlement is that, although superficially a simple problem, the analysis of the various inertial frames of reference is complex.

The twin "paradox" consists of two journeys, an outbound journey and a return journey. Much can be learnt about the relativity of simultaneity by considering just the outbound journey without any return. The single journey without any return might consist of the following scenario: Jim stays at home on Earth and Bill goes off in a spaceship, Bill flies past Jim at a velocity of 0.

Jim, who stays on Earth, finds that Bill's clocks record less time than his own for the journey. The journey to Mars consists of two inertial frames of reference. In Jim's frame of reference Bill is moving and Jim is stationary. There are only two reference frames so questions such as "what does Jim observe if he considers himself to be moving" are equivalent to asking "how does Jim move in Bill's frame of reference? Suppose for ease of calculation Mars is assumed to be light seconds away from Earth.

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Special Relativity/Simultaneity, time dilation and length contraction

Jim's view of the journey is straightforward and is shown below:. Relativistic time dilation will cause Bill's clocks to read 75 seconds ie:. In Jim's inertial frame of reference Jim has aged by seconds when Bill gets to Mars but Bill, as a result of time dilation, has only aged by 75 seconds. The reason that this lack of aging by Bill seems to be paradoxical is that it might be thought that Jim must also have aged less than Bill, after all, Jim travels away from Bill as much as Bill travels away from Jim so Bill should find Jim to have aged less than himself in the same way as Jim finds Bill to have aged less than himself.

In fact there is no paradox because Bill and Jim have very different ideas of the journey. Bill's view of the journey is shown below. As a result of relativistic phase, Bill finds that the clocks on Mars start the journey 80 seconds ahead of his own. The Martian clocks record another 45 seconds for the journey. More surprising still, when Bill considers Jim's clock readings back on Earth he finds that only 45 seconds have elapsed on Jim's earth-based clocks during the journey because the 80 seconds of relativistic phase difference mean that Bill's clocks as he passes Mars are simultaneous with events on Earth that are only 45 seconds later than when he passed Earth.

If Bill had an absurdly long rocket he could use a version of "relativistic bug capture" see above to net one of Jim's clocks the moment Bill reaches Mars. Bill could then show there was a Jim on Earth whose clocks had only progressed by 45 seconds for Bill's journey in Bill's inertial frame of reference. If Bill had captured one of Jim's clocks, the Jim who is simultaneous in his own frame of reference with the moment that Bill passes Mars would remember a giant net coming down from the tail of a spaceship and taking one of his clocks 45 seconds after Bill passed him.

Notice that there are two Jims in this story, one who is simultaneous with the Mars that Bill visits and another who is simultaneous with Bill in Bill's frame of reference. The Jim who is simultaneous with Bill in Bill's frame of reference would be an earlier Jim than the Jim who is simultaneous in his own frame of reference with Mars when Bill arrives. The one way journey is symmetrical in the sense that Jim observes Bill to age less than himself during the journey and Bill also observes an earlier Jim to have aged less than himself.

The one way journey would become fully symmetrical if Bill continued past Mars until his clocks read that seconds had elapsed, at this point he would assess that Jim had experienced 75 seconds of elapsed time. When Jim's clock's read seconds Jim finds Bill's clocks read 75 seconds and, symmetrically, when Bill's clocks read seconds Bill finds that Jim's clocks read 75 seconds. This symmetry is emphasised in the schematic diagram below which compares Jim and Bill's clock readings:. In the previous section it was shown that time dilation is symmetrical when observers separate at a constant velocity.

The symmetry is evident in the way that, during the one way journey, Jim observes Bill's clocks to go slow and Bill also observes Jim's clocks to go slow. The symmetry between the observers means that Bill can regard himself as stationary and observing Jim departing or vice versa. Bill finds that there are two Jims at two separate times involved in going to Mars then suddenly changing direction to return to Earth: these are labelled A and C in the diagram. The first Jim has clocks that read 45 seconds since Bill passed Earth, this is the Jim who is simultaneous with Bill in Bill's frame of reference as Bill passes Mars.

The second Jim has clocks that read seconds since meeting Bill, this Jim is simultaneous with Bill, in Bill's frame of reference, the moment after he has turned around at Mars and reached a velocity of The lack of any single Jim that is simultaneous with Bill when Bill changes velocity introduces an asymmetry into Special Relativity. Bill turning around at Mars to come back to Earth is not equivalent to Earth turning to meet Bill because it is not clear, until Bill has made the turn, which Earth and which Jim is making the journey.

Langevin , who first proposed the example of a traveller departing and returning younger, was well aware of the asymmetry and stated that: "Thus the asymmetry — which occurred because only the traveler, in the middle of his journey, has undergone an acceleration that changes the direction of his velocity". In Special Relativity the laws of physics are the same for each observer in an inertial frame of reference. An inertial frame of reference might be all the clocks and measuring rods in a room on board a ship or in an entire city, the crucial feature of an inertial reference frame being that the clocks and measuring rods are stationary with respect to each other.

Motions are measured so that the rest of the universe moves relative to the observer's inertial frame of reference. There are three inertial frames of reference in the example of Bill going to Mars and returning to Earth: Jim, outbound Bill and inbound Bill. Outbound and inbound Bill are separated by a period of varying velocity when Bill turns around. This period of changing velocity can also be regarded as a non-inertial hiatus in Bill's single inertial reference frame after which the relations between Bill's clocks and measuring rods and those in the rest of the universe have changed.

Questions such as "how does the Jim who is simultaneous with Bill after the turn view events if he regards himself as moving towards Bill? In Jim's own inertial frame of reference Bill just goes to Mars, turns round and comes back again. Jim always regards himself as stationary unless his reference frame becomes non-inertial by experiencing a change in velocity in which case Jim would regard himself as moving from one stationary state to another that has a different set of relationships with the universe.

Special Relativity holds that the laws of physics are the same in all inertial reference frames, it does not hold that all motion is relative, even in non-inertial changes. The gap in time between the Jim who is simultaneous with Bill as Bill reaches Mars and the Jim who is simultaneous with Bill as he starts the journey back to Earth is known as the "Time Gap". The time gap in the "twin paradox" consists of the sum of the outgoing and incoming phase differences and in this case the time gap is seconds.

Once it is accepted that Bill and Jim have very different views of the journey these views can be summarised in the "Time Gap" description of the journey. In this description Bill flies to Mars and discovers that the clocks there are reading a later time than his own clock. He turns round to fly back to Earth and realises that the relativity of simultaneity means that, for Bill, the clocks on Earth will have jumped forward and are ahead of those on Mars, yet another "time gap" appears. When Bill gets back to Earth the time gaps and time dilations mean that people on Earth have recorded more clock ticks that he did.

Zeno’s Paradoxes

For ease of calculation suppose that Bill is moving at a truly astonishing velocity of 0. The illustration below shows Jim and Bill's observations:. From Bill's viewpoint there is both a time dilation and a phase effect. It is the added factor of "phase" that explains why, although the time dilation occurs for both observers, Bill observes the same readings on Jim's clocks over the whole journey as does Jim.

Jim observes the distance as light seconds and the distant point is in his frame of reference. According to Jim it takes Bill the following time to make the journey:. So for Bill, Jim's clocks register secs have passed from the start to the distant point. This is composed of 45 secs elapsing on Jim's clocks at the turn round point plus an 80 secs time gap from the start of the journey.

Bill sees secs total time recorded on Jim's clocks over the whole journey, this is the same time as Jim observes on his own clocks. Eagle, A. A note on Dolby and Gull on radar time and the twin "paradox". American Journal of Physics. According to special relativity items such as measuring rods consist of events distributed in space and time and a three dimensional rod is the events that compose the rod at a single instant. However, from the relativity of simultaneity it is evident that two observers in relative motion will have different sets of events that are present at a given instant.

This means that two observers moving relative to each other will usually be observing measuring rods that are composed of different sets of events. If the word "rod" means the three dimensional form of the object called a rod then these two observers in relative motion observe different rods. The way that measuring rods differ between observers can be seen by using a Minkowski diagram. The area of a Minkowski diagram that corresponds to all of the events that compose an object over a period of time is known as the worldtube of the object.

It can be seen in the image below that length contraction is the result of individual observers having different sections of an object's worldtube in their present instant. It should be recalled that the longest lengths on space-time diagrams are often the shortest in reality. It is sometimes said that length contraction occurs because objects rotate into the time axis. This is actually a half truth, there is no actual rotation of a three dimensional rod, instead the observed three dimensional slice of a four dimensional rod is changed which makes it appear as if the rod has rotated into the time axis.

In special relativity it is not the rod that rotates into time, it is the observer's slice of the worldtube of the rod that rotates. There can be no doubt that the three dimensional slice of the worldtube of a rod does indeed have different lengths for relatively moving observers. The issue of whether or not the events that compose the worldtube of the rod are always existent is a matter for philosophical speculation.

Vesselin Petkov. Relativistic length agony continued. The term "time dilation" is applied to the way that observers who are moving relative to you record fewer clock ticks between events than you. In special relativity this is not due to properties of the clocks, such as their mechanisms getting heavier.

Indeed, it should not even be said that the clocks tick faster or slower because what is truly occurring is that the clocks record shorter or longer elapsed times and this recording of elapsed time is independent of the mechanism of the clocks. The differences between clock readings are due to the clocks traversing shorter or longer distances between events along an observer's path through spacetime. This can be seen most clearly by re-examining the Andromeda Paradox. Suppose Bill passes Jim at high velocity on the way to Mars.

Jim has previously synchronised the clocks on Mars with his Earth clocks but for Bill the Martian clocks read times well in advance of Jim's. This means that Bill has a head start because his present instant contains what Jim considers to be the Martian future. Jim observes that Bill travels through both space and time and expresses this observation by saying that Bill's clocks recorded fewer ticks than his own.

Bill achieves this strange time travel by having what Jim considers to be the future of distant objects in his present moment. Bill is literally travelling into future parts of Jim's frame of reference. In special relativity time dilation and length contraction are not material effects, they are physical effects due to travel within a four dimensional spacetime.

The mechanisms of the clocks and the structures of measuring rods are irrelevant. It is important for advanced students to be aware that special relativity and General Relativity differ about the nature of spacetime. General Relativity, in the form championed by Einstein, avoids the idea of extended space and time and is what is known as a "relationalist" theory of physics. Special relativity, on the other hand, is a theory where extended spacetime is pre-eminent.

The Arkansas paradox « Statistical Modeling, Causal Inference, and Social Science

The brilliant flowering of physical theory in the early twentieth century has tended to obscure this difference because, within a decade, special relativity had been subsumed within General Relativity. The interpretation of special relativity that is presented here should be learnt before advancing to more advanced interpretations. The length contraction in relativity is symmetrical.

When two observers in relative motion pass each other they both measure a contraction of length. The symmetry of length contraction leads to two questions. The other thing I find interesting about this puzzle is wondering whether the people who find the arguments in favor of switching convincing are actually reasoning better than I am. If so, how are they doing this? What is it that makes those allegedly superior arguments convincing to them? The thing that kind of convinced me that the pro-switch arguments must be correct is that people have run simulations of this problem on computers and the switching strategy turns out to win in line with the pro-switch argument probabilities.

But the tricky thing about probabilities and randomness is that such successes could be "just a matter of luck", as it were. Maybe they were wrong, but just got lucky. It's improbable, but still possible. Only mathematical proof independent of experimentation should be rock-solid correct.

But mathematical proof seems to be a matter of being convincing enough to mathematicians, and no so independent after all, and that seems disappointing somehow. There's a feeling that mathematical proof should be above mere argumentative skill and the audience's susceptibility to being convinced. But there is only one car, so the set of doors you didn't pick must include at least one non-car. The host knows the location of the car so can always open a non-car door from this set. By swapping you can effectively think of it as choosing both other doors.

An easy way to get your head around this is increase the number of doors, say to You pick 1 of those , the host opens 98 other doors, do you swap? Alternate reality rules of the game are: nobody speaks, you pick any two doors you like, and you win the prize behind both of them. Monty will gesture at you to point to a door. You point at the one you don't like. Monty will then show you one of your prizes behind one of your two chosen doors.

Monty will then gesture for you to point to a door. You point at the remaining one you liked. Monty then gives you your second prize behind that door. Notice that all of the externally visible actions are identical to the original game. I don't get this. Why does't the player insist that they open door no 1. Or is it part of the game that the host will open a different door other than the one the player picked If the host knows what is behind the doors, and you are only given two chances, and the host can open a different door from what you picked, isn't your chance of winning zero?

I don't get this Retra on Apr 6, The host's job is to create suspense by opening a door and offering a chance to switch. If they opened the door you picked first, there would be no suspense, and if they didn't know what was behind the door, they could show you that you lose, which also removes suspense. Turns out the host just ends up giving you more information that you can use to make a better decision. Can you please explain this.. I love this one, but I guess it must be less common to have such a problem in the real life. Or I'm wrong? The best explanation on the solution for this problem is from Numberphile.

The Fermi Paradox — Where Are All The Aliens? (1/2)

Properly applying Bayesian inference gives you the correct answer. Also the answer becomes intuitive once this scales to more than 3 doors someone below already mentioned. When we discussed this in school I quickly wrote a program that simulated it No need to look at fancy paradoxes, just think about the following. If you think you know the answer, you are probably wrong. EDIT: Instead of just voting this down, try to give an answer. If you think it is easy, you have not thought about it careful enough.

of time volume Ebook

Except for the risk-neutrality detail this is all Probability , right? Or are you thinking of something else? You would have to be exceptionally lucky, but you could get heads, heads, tails repeated for ever and therefore the relative frequency of heads would fluctuate increasingly tiny amounts around This of course has probability zero, but it is not impossible.

The Bayesian view is problematic for several reasons. Why do I need someone with believes about the coin, we are talking about intrinsic properties of tossing a coin after all. And if that is not enough, we also throw some betting in. Tossing a coin does certainly not depends on the invention of money and gambling, at least ignoring that coins are usually money. You would have to be exceptionally lucky, but you could get heads, heads, tails repeated for ever This would violate the law of large numbers.

You may end up with the sequence HHT 10million times, but the chances that you continue to get that sequence for another 10million times is all but zero, and then gets even smaller as you add another 10million trials. No, the law of large numbers does not assert that all sequences of outcomes converge, only that this happens almost surely. Or look at it the other way round, what mechanism would prevent heads, heads, tails repeated forever? I can certainly get heads, heads, tails on the first three tosses. After that I start over, three more tosses all independent of what just happened, again a Why could this not continue forever?

I'm pretty sure your confusion about probability stems from you not understanding the mathematical concept of a "limit". Your sequence HHT has a No, I will get similar counts with high probability but not surely. There are sequences of probability zero that do not converge.

Think about it this way. Every coin toss in a sequence, finite or infinite, on its own can surly turn out to be heads, can't it? And all tosses are independent, aren't they? So why can't all tosses turn out heads? Unless you have a convincing argument why some tosses have to yield tails eventually, you have to deal with the fact that not all sequences of tosses converge to the expected probability. I would agree if the sentence contained just the word "finite". The "or infinite" is where you are thinking too intuitively, and not mathematically correct anymore. The difference between "finite" and "infinite" is precisely the solution to this paradox in your mind.

I hope to be able to point out that it is easy to correct for that with just a bit of structured but maybe non-intuitive thinking. One of the simplest and I would argue most complete definitions of "a probability of 0. Think about this for a second, and I recommend to also use this opportunity to appreciate again that half of infinity is still infinity.

This relates the mathematical concept of infinity to the definition of probability. With this definition, it probably feels like I so far just reworded your question. That may not be satisfying. So, I would like to encourage you to imagine that you have superpowers and can actually perform an infinite number of tests. You do that on a sunny day and observe that all test outcomes were the same: A. You call it a day and you can conclude using the mathematical definition from above in your diary of days-with-superpowers: "Today I have empirically determined that the test shows outcome A with a probability of 1".

You might smile and add "Peter said that outcome A has a probability of 0. In other words: if you do an infinite number of tests, the normalized distribution of test results precisely is the probability distribution of test results. I think we have learned by now that the concepts of infinity and probability are deeply related and can, by definition, be used to explain each other.

That might still not be satisfying. So, I would like to focus on the "finite" case for a bit. Imagine you don't have super powers anymore, but you're pretty resilient and motivated and you want to do the experiment to in validate Peter's claim: "The probability for both, outcome A and B, is 0. After 1. You're tired from all the testing and you complain correctly! I am pretty damn sure that he is wrong! How long do I still need to do this to be absolutely sure? Infinity usually does not allow for actual intuitive thinking.

But there are a few really simple mathematical rules around infinity and convergence that make it actually pretty simple and again intuitive to deal with the concept. I appreciate your attempt but you did not convince me the slightest bit. Let's take the 1,, coin tosses all heads. This result has no bearing on the probability of the coin at all. It may make you strongly doubt that the coin is indeed a fair coin but - and that is the point I am trying to get at - there is nothing that prevents a fair coin from coming up heads 1,, times in a row.

Whatever your experiment shows, it could always be a statistical fluke. And the infinite case does not changes much, at least not in a way obvious to me. That was fun, let me do that again tomorrow. This is the way it goes for a long time but then something strange happens, one day all tosses come up heads. The very next day everything is back to normal. What is now the probability of heads, we got two different answers for your way of defining he probability? And all it took was an extreme statistical outlier on single day.

The point of probability is that performing an experiment an infinite number of times guarantees that every outcome happens with a proportion that exactly equals its probability for a formalization of what that even means, look at measure theory. If you get different proportions on different days, you have different probabilities. That means, you weren't performing the same experiment. Not OP but I think you still don't grasp the concept of infinity. Terms like "the very next day" or any other segmentation don't apply.

Its always good to be in such a business except for they cant be proven right either in finite time. I agree, there are problems and non-intuitive aspects to both approaches to probability. Otherwise there wouldn't be two approaches. I think your objection to frequentism unfairly conflates the idealized world with the physical world. Also, you left out 2 arguments against frequentism: it allows inconsistent beliefs, and in practice it has allowed bad approaches in scientific papers. As for the Bayseian view, being non-intuitive isn't the same as problematic.

And this seems to force you into an infinite regress. But not quite, in very rare cases it won't. But now you have to quantify that this tiny fraction is something of probability zero. And even worse, it is still possible that none of your repeated experiments showed convergence, you seem right back where you started. I would love to know to what you are referring with the inconsistent beliefs. I would not say that Bayesian view is non-intuitive, I would say it fails to account for important things.

A priori probabilities have to be rooted somewhere. Because you have previously observed the relative frequencies of coin tosses? Because you made some theoretical observations about symmetries?

The Arkansas paradox

There must be, at least so it seems to me, something about the probabilities associated with coin tosses that is independent of any individual, otherwise it would become rather difficult to explain how different individuals would arrive at similar probabilities independent of each other. So banning probabilities into the realm of beliefs does not cut it in my opinion.

You could build frequentist models for the odds of 0 to 1 inches of rain falling in Cleveland tomorrow, 1 to 2 inches, and 0 to 2 inches, and the odds of 0 to 1 plus 1 to 2 don't need to add up to 0 to 2 inches. You can justify all 3 models, but they are inconsistent.

Bayesian models don't allow that. Here's an online source asserting "frequentists can have two different unbiased estimators under the same likelihood functions. Infinity isn't a number; you can't multiply it by another number and get a meaningful result. You are saying you could run an infinite number of tests on Monday and then run an infinite number of tests on Tuesday and so on, but the concept of infinity holds that you did as many trials on Monday as you did on Monday and Tuesday combined.

If you want to go down this road, then we will have to switch to a nuclear decay based coin or something like that. In the case of a coin toss there never was any real randomness, as you say we were just ignorant of the initial conditions. Given a distribution over the possible initial conditions, we can determine the probabilities for heads and tails.

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The possibly huge number of degrees of freedom and deterministic chaos will of course make this an unpleasant exercise. Can we base our belief about coin toss probability on the belief that about half of the initial conditions lead to heads and half to tails, without verifying it?

So, what's the right answer? If I knew, I wouldn't have asked.

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And as far as I can tell nobody knows or at least there is no commonly accepted answer. I hope to have convinced with my previous answer that the mathematical definition of probability is key to resolving your paradox here. Mathematically using the concept of infinity and also physically using the concept of large numbers there is a simple and commonly accepted answer to your question.

Since that definition is very basic, it is hard to find a direct answer to your question. Logical consistency and a certainty that is large enough are usually key to success i. I guess another way of stating the problem is this: say you toss a coin and get heads the first times. Now you are likely to believe that the coin is biased. But that cannot be proved. It may be that you have simply not tossed the coin enough times to perceive its fairness.

Maybe after the th toss you start to get enough tails such that after the millionth toss, it is not at all clear that there is any bias. So practically we can make a judgement as to whether the coin is biased on not based on how many tosses we think is sufficient, but theoretically it is impossible to distinguish a biased coin from a fair one if we toss to infinity. Yes, that is exactly what I had in mind. Probabilities are weired in the way that they say what will happen but then still leave open the possibility that this will not happen at all.

You toss a coin a billion times, you will get about million heads. Well, or you don't and there are exactly zero heads. TekMol on Apr 5, You can play this game for every statement. What does it mean that an elephant is bigger then a dog? That one is easy. First let's clarify what exactly we mean with bigger. Let's say we mean larger in volume. Now we pick a procedure for determining the volume of animals, say we submerge them briefly in a large water tank and mark the resulting water level.

That animal that lead to the highest water level is the bigger one. Any objections? TekMol on Apr 6, The same objections you have regarding the fair coin. You say you submerge the animals in a tank of water. How does that work? You have this giant tank, put in the animal and one molecule of water?

Just like a single toin coss, a single molecule of water will not tell you much. You need hundreds of billions of water molecules? And how do you know these will behave in the expected way? Because you have a model in your head how molecules behave? Well, I have a model how fair coins behave. Because you saw different sized objects result in different water levels before? I saw fair and unfair coins result in different head:tail distributions before.

But similar to your objection about the Bayesian view of probability requiring money and betting to be defined, you've now defined the idea of size to be dependent on giant tanks of water. That we're modeling the outcome of a coin toss as a sample space set containing at least two event elements, one of which we call "heads," and that we have a probability measure which assigns 0. You are presupposing an understanding of probability, aren't you? What does it mean that a probability measure assigns 0.

Yes, but what are the consequences of this definition? The probability has no implications about the coin. It has implications about how you make decisions concerning the flipping of coins. It almost sounds to me as if he is pushing into gambler's fallacy space. Like if a fair roulette wheel hasn't hit 7 in spins, then 7 is somehow due. No, his arguments is exactly opposite to the gambler's fallacy. If the gambler's fallacy were not a fallacy, then there would be no issue.

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But because we know it is a fallacy, then the fact that we have heads so far tells us in theory nothing about the fairness of the coin. If I flip a coin times and always get heads, how much would you be willing to bet me that the next flip gets tails? So after how many consecutive heads will you decide that the coin is biased?