![[NEW VIDEO!] The Gambler's Fallacy is Really Odd](https://img5.xaiju.com/storage/9/lr/xh/d38796-019e7bd1-2d17-7417-a027-58095e5d119c.jpg)
[NEW VIDEO!] The Gambler's Fallacy is Really Odd
Added 2022-08-24 19:37:33 +0000 UTCSo, did you fall for it? ;)
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The first half of my post above has vanished. I'll insert it here so you'll have the context. I'll check on its half-life in a day or two. A mechanical pachinko machine from Japan releases a ball at the top of a triangle and lets it fall upon a peg that gives the ball an equal chance to recoil to the left (x-option) or the right (y-option). The ball falls again and strikes a pin that splits the ball's path into two equally weighed options. An electro-mechanical machine can use half-silvered mirrors or calcite crystals for beam splitters and become a network of Mach-Zehnder interferometers. As photons zig-zag down from the top vertex, they map out the geometry of the pachinko machine and reveal any biases that might be in the fifty-fifty splits at each half-silvered mirror. After a thousands descents and forks in the road, where would one expect to find a solid ball or the most illumination? Most likely, close to the middle of the thousandth row. If after many repeats of the experiment the ball tends to one side of the triangle, one can suspect that the machine has a tilt or a defect in it. In Newton's time, Blaise Pascal realized that each repetition of a chance event creates a new row of high order terms as surely as when one multiplies a binomial (x+y) times itself. For each power of (x+y)^n , Pascal wrote out the coefficients row by row, starting with a "1" for n=0 at the top. Pascal visualized how with the nth toss of a coin, one effectively multiplies a binomial (x+y) times itself n times. He wrote the coefficients for (x+y) raised to the zeroth, first, second, third powers as four neat rows of numbers centered below the top vertex. One can continue the triangle downward and include higher powers of (x+y)^n . With each iteration, the next row down forms the next wider tier of the triangle. The row number matches the exponent n for how many times the binomial gets multiplied times itself. Each coefficient in the triangle equals the sum of the two closest entries in the row above. Each row has its numbers getting larger as one moves in from the edges to the middle. The coefficients in the nth row add up to 2^n. When you make one of these triangles, you start with a number one at the top vertex for (x+y)^0 = 1 = I = Identity matrix for do nothing, no change.. An uninterrupted string of ones runs along the sides of the triangle. Just inside the smooth edges, one has the outliers with just one bend in their downward path down. Each number in Pascal's triangle gives the exact count for how many paths can be threaded up through the coefficients to the vertex at top. If one starts along one of the slanted edges, one has only one way up. Deep in the bulk, one has many ways to wiggle one's way to the top. After thousands of test runs, one can see if the row numbers create the symmetric ideal triangle one gets from pure mathematics. The ideal never gets attained, however, it serves as a model that one can approach asymptotically yet never touch. Particles moving through a pachinko machine can spread out to a maximum width and then reflect off the outer walls and gather together in a single hole at the reflected bottom of the machine. One has a model here of of water entering a tank at the top and flowing out. In a spacetime diagram, the double Pascal triangle represents a causal diamond. The top and bottom vertices represent places a quantum particle has visited. The numbers in the double Pascal triangle tell any intrepid traveler at that position how many ways from there lead to a vertex. The central axis from one vertex to another acts as an attractor of all of the larger numbers. A straight path from top to bottom through the largest numbers automatically satisfies a principle of least action, because the direct path has the greatest number of ways of forming and being completed. A first order deviation away from the center line causes at most a second order change in the nearby coefficients. As one moves down Pascal's triangle, the numbers in the middle become ever larger compared to those along the edges. The numbers in each row reveal how many ways there are to weave one's way to a vertex. When one reflects Pascal's triangle on a row and creates a causal diamond in spacetime, the vertices represent events in which a particle's location happens to be known. In between, in the bulk of the diamond, all sorts of virtual processes can unfold and come back together so that a straight looking path, when zoomed into a fantastic degree gets a pattern like ----<>------<< >>-----<>-----. With the help of computers, Pascal's triangle has been displayed for expansions with rows a thousand terms wide. A self-similar fractal like architecture becomes apparent showing what Carl Sagan called the lumpiness in randomness. In the hands of Stephen Wolfram, Pascal's triangle becomes one of many cellular-automata to consider, each with its signature lumpiness. When playing with a large numbers of rows, or even just certain selected rows or diagonals, one can highlight curious weaves in nature's loom. Long before the advent of computers, mathematicians figured out the coefficients for every row of Pascal's triangle. At the infinite limit for the widest imaginable row with an infinite number of terms of the highest order, we have a smooth bell-shaped curve named after Johann Carl Friedrich Gauss. If one (conformally) compresses Pascals triangle and brings the edges of its infinite row within view, if one squeezes it all in enough to show Gauss's infinitely wide row at the bottom, then one gets a Dirac Delta function for a quantum particle _|_ Gabriel's horn. The juxtaposition (x+y) can involve any pair of objects of the same rank added or superimposed together. The x and y can be numbers, vectors, functions, even matrices (keeping xyx separate from yxx, if x and y do not commute). One can let x=1 or be a reference point or an initial point that we indicate in higher dimensions as an Identity matrix I.
Scott Ready
2022-09-03 19:13:26 +0000 UTCA resolution of the Identity matrix doesn't display correctly here . . . An Identity matrix can be resolved many ways depending on one's basis. A simple looking Identity matrix takes on new interest when one considers its square roots, Pauli matrices and Dirac matrices. Let y be a tiny shift away from an Initial Condition I. The binomial expansion of ( I + y )^n where y behaves as a differential gives one a gateway into calculus. Take it to the limit, to the widest row and you'll find that a first order shift away from the vertical center-line of Pascal's triangle causes at most a second order change in the coefficients. The spinning wheels of a casino behave as smoke and mirrors, misdirection and eye candy. Flashy wheels and dice mesmerize one's attention, and yet, as Feynman observed, the number of whole revolutions of the wheels makes no difference in the outcome -- just the last digits on the odometer. What counts can only be the final partial movement of the spinning object, its phase upon detection. The number of free revolutions beforehand does not get remembered. A quantum amplitude has a phase angle at impact that gives no information as to how many times a particle's clock hand might have spun around beforehand. All of this mathematics pouring out of a single triangle does not just get invented; it gets discovered. At the heart of the Scientific Revolution we have an ideal called repeatable experiments. Just because an ideal can't be reached, it still serves a purpose in being universally understood. With the help of four generations of Bernoulli's and Pascal, each run of an experiment gets treated as an independent and repeatable test run, a Bernoulli trial. The scientific method rests on repeatable experiments. The partial differentials of calculus lie at the heart of controlled experiments in which most variables are held fixed while just one or two or a constraint of several varies a little. No one would want to repeat a pandemic on Earth ten times just to build up some statistics. Neither a biologist nor a volcanologist can expect to have exactly repeated events. A volcanologist can shift to another continent or another planet or moon and study "volcanoes" under altered conditions. We can test and reset for tiny changes, differential shifts. When a great many repeatable components get iterated and networked into a large system, global features and surprises can occur that one cannot compute from simple gates. The computation of the network outpaces any local computation. Therefore, no one understands their health bill or the cost of anything hugely public. In the words of Max Planck* ". . . measurements give no direct information about external reality. They are only a register or representation of reactions to physical phenomena. As such they contain no explicit information and have to be interpreted. As Helmholtz said, measurements furnish the physicist with a sign which he must interpret." "... Taking it, then, that the external world of reality is governed by a system of laws, the physicist now constructs a synthesis of concepts and theorems . . . a representation of the real world itself in so far as it corresponds as closely as possible to the information which the research measurements have supplied. Once he has accomplished this the researcher can assert, without having to fear the contradiction of facts, that he has discovered one side of the outer world of reality, though of course he can never logically demonstrate the truth of the assertion." "From the positivist standpoint, of course, this idea of constructing a scientific picture of the physical universe— this continual striving after a knowledge of external reality — is something foreign and meaningless. For where there is no outer object there is nothing that can be portrayed or described." "Every applied hypothesis which succeeds in throwing the searchlight of a new vision across the field of physical science represents a plunge into the darkness; because we cannot at first reduce the vision to a logical statement." *Where is Science going? by Max Planck, Prologue by Albert Einstein, 1932 English translation here by James Murphy.
Scott Ready
2022-08-31 21:54:30 +0000 UTCNice video! I was wondering if you were going to cover how sampling is done for stuff like surveys and polls, in order to get a reasonable estimate of what's happening in the general population. Seems like a related topic, though maybe not a rabbit hole you really want to get into here. :)
Andrew Pann
2022-08-24 20:14:13 +0000 UTCI already watched the video, if that's what you mean. And fell for it quite hard, as usual
Stefan Nikolov
2022-08-24 19:40:41 +0000 UTC
