if length X has uniform distribution, X^2 does NOT have uniform distribution. conversely if you take X to be area, then sqrt(X) doesn't have uniform distribution.
I fail to see the paradox here. In one case you are assuming the lengths are uniformly distributed and in the other that the areas are uniformly distributed. All this shows is that you should think more carefully before assuming a distribution pattern when making a first guess.
Yeah notice however that in this range x^2 (or sqrt(x) for that matter) is a bijection, thus it is plausible to assume that if the length is uniformly distributed, so is the area. What you need to show is, why it is fallacious to assume that the area is also uniformly distributed, to which I have no answer atm.
The problem isn't that it is fallacious to assume a uniformly distribution for the area OR the length, but it is easy to demonstrate that these two assumptions are not compatible with each other and therefore you would EXPECT different answers based on which you chose.
I also noticed that via computer simulation. Choosing a number in [1,3] and squaring it gives something like an exponential (or maybe 1/x^q) kind of distribution. The question is *why* that is the case
The real paradox here is the following: The principle of indifference says you should treat all possibilities as equally likely if you don't know anything about them. But there often seems to be _no unique way_ to divide the possibility space into possibilities! In combination with the principle of indifference this leads to multiple inconsistent probability distributions. In my opinion this shows that the principle of indifference doesn't provide an "ignorance prior" for Bayesian reasoning. But what then is the right principle to represent ignorance? My guess is that a version of Ockham's razor could be the solution, but this is very hard to prove.
You can't/shouldn't chose a a "starting point" for choosing a probability based upon the range of _possibilities_. If you don't know the probability distribution, you don't know the probability distribution, and there is no way around that fact. If you only know that a factory produces squares of wood with sides between 1 and 3 feet, you don't know the distribution. The factory could produce all 1 foot squares for all you know. Or it could produce all 3 foot squares. The information that the squares are between 1 and 3 feet just doesn't tell us anything about probabilities.
"Randomly" is not a probability distribution. Suppose I have a bag that contains red and black marbles. If I pick a marble from the bag randomly, it means that I pick without any knowledge of which marble I might choose. However the probability that the marble I pick is red depends upon the distribution of red and black marbles in the bag. If there are 100 red marbles, and 1 black, then if I choose a marble randomly, it is 100 times more likely to be red than black. Asserting that I choose randomly is not the same as asserting that there is an equal probability of choosing a red or black marble.
" You can't/shouldn't " I agree that "you can't" however when you say "you shouldn't" I think you're completely detached from reality. Most of human knowledge (at least in our modern society) is at some point based on some kind of dubious "probabilistic" reasoning. When your doctor say XXX pregnancy test is accurate at 95%, do you believe this is a "true" probability they got from some "real" probability distribution? In the example showcased in the video, they began with the "length" description of the problem then pointed out there was an other description (the "area" one) that contradicted the first one. In real life situations that "area" description is often hidden. You rarely/never "know" the probability distribution a certain phenomenon follows and even if you end up with a distribution you believe to be correct then you still won't know what is the probability you are wrong: you never know if there is some kind of hidden "area description" that would destroy your reasoning/belief.
When a doctor says a test is accurate 95% of the time, that is because the test "has been tested". However, if a test has two possible outcomes, and it hasn't " been tested" there is no reason to believe a priori that the test is accurate 50% of the time. Should we make judgments based upon probability? Absolutely. Should we assume, without further information, that if there are x possible outcomes of some activity that each outcome has a probability of 1/x? No, that would be foolish. Now there might be _evidence_ that the probabilities are equally distributed, (or not), but in the absence of evidence, there is no reason to assume something that is very very often not the case.
You are entirely focused on the exact formulation of the problem presented in this video and totally miss the big picture. Of course, in this specific problem the obvious answer is to mass produce the cardboards then look at the statistics or simply walk into the factory and look at how they are made... Or better: since there is nothing at stake here, I guess that ignoring the question and saying "I don't know and I don't care" is also a possibility (which seems to be your stance here). But you should really imagine a real life problem where that kind of easy exit is unavailable... Again, when you say the real issue is the fact the uniform distribution rarely appears (which I disagree with) or that you don't have any evidence about anything in this video you totally focus on the exact formulation of the problem showcased here. The kind of problematic showcased in the video doesn't only arise if you take the uniform distribution as your prior, also, the problematic still remains if you have a certain amount of evidence that you think is leaning toward one answer or maybe you know 2 or more answers but think you can rely on Occam's razor etc. Lastly the fact that you put quotation mark around "has been tested" means you know something weird is happening for the "test", which is the whole point of my argument.
Random Name Probability is indeed a well-defined area in mathematics! This is, however, not a question about the mathematics so much as it is a question of the interpretation of those mathematics. There are, in fact, a number of conflicting ideas regarding the interpretation of probability-what it is how it relates to chance and frequency and all of that-held by different mathematicians and philosophers. One that used to be popular was the Classical Interpretation, which first appeared in the work of Pierre-Simon Laplace, defining probability as simply the ratio of favorable outcomes to the total number of equally possible outcomes. This interpretation hinges on the ambiguous term “equally possible.” On first read, it seems identical to “equally probable,” which would make it circular as a definition of probability. To clarify that and get rid of any circularity, La Place introduced what John Maynard Keynes eventually labeled “The Principle of Indifference”: in the absence of any evidence inclining you to any particular outcome, assume a uniform distribution of the outcomes. What Bertrand’s Paradoxes show us is that the Principle of Indifference is deeply and perhaps irredeemably flawed as an assumption, as it can lead us as easily to assume multiple different distributions depending on how we present the same questions. Since contradictions follow from the Principle in these paradoxes, we are forced to abandon the Principle, at least as a part of an objective general theory, and look to other non-Classical interpretations that do not depend on it, like those developed by frequentists (Von Mises), propensity theorists (Popper, Keynes, Carnap), and Bayesians (Fintetti). That doesn’t, however, mean that the Classical Interpretation doesn’t work to help clarify our thinking-especially regarding simple, discreet problems-and can’t be useful. As long as you make clear which distribution you’re using, you should be fine in most applied mathematical/statistical situations. But, still, as a general approach, Bertrand’s Paradoxes killed the Classical Interpretation. (I’d very strongly recommend reading the Stanford Encyclopedia of Philosophy article on interpretations of probability, then checking out some of the literature it cites, or perhaps picking up a copy of Timothy Childers’s wonderful “Philosophy and Probability” from Oxford University Press.)
Neither answer is "right." The 1/2 assumes that length is uniformly distributed, and 3/8 assumes that area is uniformly distributed. These are different assumptions! It's no surprise that different assumptions lead to different conclusions! That's not a paradox. In the real world, you would measure different pieces of timber and find a probability distribution that best fits the data. Then you could use your model to predict what properties future timber will have. Without data, all you have is a guess, based on assumptions. Your assumptions may be right or wrong, but you can't know either way until you have data.
it is not a paradox if you just state explicitly that "we know there will be no squares of area 5 6 7 and 8), so before we lay things down on an axis we have to look at how is the set (of square areas) distributed and choose the appropriate axis to represent them. In this case a linear axis is wrong, what do you think would be the counter argument to this?
For those mentioning an error in the assumptions about taking length as uniform and then the area is not uniform, there actually is no problem here. Because the set of all possible lengths of squares between 1-3 can be isomorphically (and hence injectively) mapped to the set of all possible areas of squares between 1-9 (since we define squares to have positive length). So the space of length probabilities has the same structure as the space of area probabilities. So yes there is a real paradox here
@luizhenriquedefreitasassis2051hi, your first question about the number of real numbers between a given interval is somewhat ill defined, since any interval in R is uncountable. What I think people are trying to say is that, asking the question “what is the probability of length being 1-3?” is not the same as “what is the probability of area 1-9”. But they are the same, since for each square with area A between 1-9 there exists a unique length L between 1-3 such that L^2 is the area. So if these sets are isomorphic under y=x^2 then the two questions must be the same
Sticking my head out and I am aware I may be absolutely wrong, but I think this approach may solve the paradox: In my most humble opinion, the paradox arises due to, for instance, comparing apples with oranges, since we are comparing values in a linear range with values of a quadratic function. To compare like with like, think slope (1st derivative) of the quadratic curve for corresponding area values should be used. The 1st derivative (slope) of the equation y = x^2 can be calculated as 2x. The corresponding values at relevant intervals can be calculated as follows: for length x = 1, area = 1, and 2x = 2; for length x = 2, area = 4, and 2x = 4; for length x = 3, area = 9, and 2x = 6; A: corresponding 2x range for interval x = 1 to 2, is 4 - 2 = 2; B: corresponding 2x range for interval x = 1 to 3, is 6 - 2 = 4; Therefore, corresponding probability of A / B can be calculated as 2 / 4 = 1 / 2 = same as the probability calculated for corresponding length range.
Naresh K This would also work if you took the second derivative of volume and plugged in the same values. I think you’re on to something, but I’m having trouble assenting with the derivation from analogous functions to bring apples back in comparison with apples. Taking the derivative like you have verbally yields the rate of change of area with respect to the length of a side. After your derivation, units become unidimensional for comparison, but it’s dissimilar to deriving velocity with respect to time to get distance.
I don't get it. The correct answer to the questions at the beginning is "I don't know because I have no way of knowing or even estimating the probability distribution." Can someone explain Bertrand's paradox starting from here?
Penny Lane There really is no paradox. It's just a reductio ad absurdum of the classical interpretation of probability (and hence the principle of indifference).
Benjamin Przybocki It applies to subjective probability (degree of belief) just as well. What principle can we use to represent ignorance/undecidedness? An obvious answer: The principle of indifference! It says: If you don't know anything about how likely something is, just treat all possibilities as equally likely. Sounds pretty rational, eh? But there is no unique way to divide a case into possibilities, which results in the principle of indifference giving you inconsistent probability distributions! So if the principle of indifference is no good, are we irrational when we rely on it!? And what other principle _does_ represent ignorance, if the principle of indifference doesn't do it?
Using area implies we’re not talking about squares only. We’re talking now about rectangles too. So either there is no paradox(high chance for this as I don’t think there exists such a thing) or you just did not explain the paradox correctly.
The problem was described in terms of the length 1-3, not the area 1-9. The distributions of probability do not match between these two, as the second will scale as the first squared. So you must chose only one. If you assume an even distribution for the length (which is not stated, but assumed because the problem was described in terms of length) then the probability is 1/2. The paradox is only present because the probability distribution is omitted, so it's not a paradox, merely an incomplete problem.
Exactly what I was thinking. The crucial thing is: what is the factory doing? Making random lengths with equal distribution or random area's with equal distribution? That's not the same thing. Nothing paradoxical happening here.
I mostly agree with you. However, I disagree that just because the range of sizes is presented in lengths, that we should assume that lengths rather than areas are uniformly distributed.
It actually doesn’t matter what the underlying distributions are because if you know one (for example the distribution of the lengths) you get the other (area) distribution for free since it’s the same (for the set of lengths is isomorphic to the set of areas)
The author of the video made an error when he decided to not talk about the original Bertrand's paradox. It's not complex at all and I am sure that many viewers who are clueless after watching this video wouldn't have any problems understanding the original Bertrand's paradox. The original is a very interesting thought experiment which indeed has had influence on science. This video however is objectively incorrect, because the author incorrectly tried to find a different analogy. The answer 3/8 is objectively incorrect. All the commentators claiming otherwise have no clue about mathematics or Bertrand's paradox. I will explain why. This video clearly states that the shapes produced are squares. Squares have only one independent variable, let's call it A. The problem presented implies that indeed this A is the variable that follows uniform distribution, not the AxA. When the second part of the problem was presented it was clearly talking about sides still being restricted to same values for A. Therefore we need to use only variable A when we are looking for probabilities. No mathematician in the world would get an answer 3/8 to this problem, not the way the problem was presented. Every decent mathematician would transform from the area to the sides and get the correct answer 1/2. Is it possible to get answer 3/8 and for that answer to be correct? Yes it is, but the problem would have to be presented differently. Instead of saying that the factory manufactures squares, the problem could claim that the products are completely random shapes, they don't even have to have straight edges, completely random. In that case you don't know absolutely anything about the edges, angles, convexity etc. Therefore you would have to assume that the areas of the products are uniformly distributed.
You say, "squares have only one independent variable, and it's side length." I could respond, "squares have only one independent variable, and it's area." To infer that because the minimum and maximum sizes are given in terms of side length, that side length is uniformly distributed, is a non-sequitur, and I suspect that very few mathematicians would make that mistake.
In fact, I think the only correct response to the question, "what is the probability that the side length of the next square is between 1 and 2?" is to ask, "Well, what's the probability distribution of side lengths?" As far as I know, uniform distribution is not a default assumption in any area of mathematics or science.
This is what has irritated me in some probability problems: they ask you to solve probabilities without giving the underlying probability distribution. This leads to misunderstandings like presented in the video. And I think (or hope) scientists also are aware of this - that's why we have Bayesian statistics.
This is called "the problem of priors". As a Bayesian you can't always expect somebody to give you "the underlying probability distribution". You have to pick it for yourself, subjectively. Or with help from some principles like the principle of indifference or the principle of maximum entropy.
Sarastro404 The real paradox here is the following: The principle of indifference says you should treat all possibilities as equally likely if you don't know anything about them. But there often seems to be _no unique way_ to divide the possibility space into possibilities! In combination with the principle of indifference this leads to multiple inconsistent probability distributions. In my opinion this shows that the principle of indifference doesn't provide an "ignorance prior" for Bayesian reasoning. But what then is the right principle to represent ignorance? My guess is that a version of Ockham's razor could be the solution, but this is very hard to prove.
The trick is when you're looking at the lengths you're assuming a linear distribution of possibilities and when you're looking at the area you're assuming an exponential distribution
Jonathan Weisberg, the UoT philosophy professor who called the 9/11 Commemoration a "dumb little annual pity party?" Philosophy didn't offer you a moral spine, I see.
As a statistician, I have to admit I don't find the example really convincing. If L is uniformly distributed between 1 and 3, it will be less than 2 with probability 0.5. Similarly, the area A=L^2 (which is *not* uniformly distributed) will be less than 4 with the same probability 0.5.
I liked it. Using a more complex (realistic) example would have made the problem seem less like a simple misunderstanding, but it would also have pulled focus away from the core of the problem. Nicely done!
In this case, it looks infinity shouldn't have been applied to real world length in the first place. It's paradoxical to assume there are infinite fractions(space) in a finite length.
Not a mathematician here, but this doesn't seem to be a paradox. Length and area are not going to function the same way in the same equation, because they are not measuring the same things. It is inconsistent to say we're looking at the distribution of squares with an area of 1-2 ft^2 without recognizing that we are then fundamentally doing a different thing from judging the distribution of squares with a side length of 1-2 ft. In order for this to be a paradox, the distribution of area among the squares would have to be gauged by the same criteria as distribution via side length, and still not work, so "How many squares will have less than 4.5 ft^2," as 4.5 is half of the total maximum area of 9 ft^2. This sort of silliness is largely what kept me from entering math as a field. Math is a language of logical description, but Bertrand's Paradox is a great example of how communication can break down regardless of tongue. ;) Great explanation and video! Looks like you started some spirited conversation in the comments!
It seems to me a paradox involves some difficult tension -- incompatible commonsense intuitions or contradictory provables -- forcing us to make a hard choice. But there is no hard choice here, no resolution required.
Is there a time in real life when you don't lack data? Mathematically there is no real paradox but as pointed out in the video the problem was about real life situation (and scientific knowledge).
There is, if you lack data, don't rush to conclusion, take the time to gather more. There's a lesson there; Assumption often leads to wrong conclusion.
In principle you are right but in practice that's not how we, humans, did to get where we are now. To gather data efficiently you often start from assumptions because they serve as basis for how and where you search for (relevant) data. Then you sometimes still need to interpret the data and this is also where your prior assumptions can be needed (example: quarks, Higgs bosons are not "seen", they are interpreted from some signals but that interpretation is partly based on the very theory physicists were trying to prove/disprove when trying to detect them). Historically, theories are built through trials and errors: you assume stuffs that seem reasonable, wait and see if some discrepancies appear then refine your assumptions if needed. (and I'm not just refering to science, I heard babies use some succesive bayesian inferences to learn a language.)
You only get to pick one linear distribution. If the edge has a linear probability, the P(50) area is 4. If the area distribution is linear, then the P(50) edge length is the square root of 5. There is no paradox.
the given information defines the problem not the post problem semantics. there is a problem/question/puzzle or there is not. problem is length of side NOT area qed: 1/2.
The first question, 1 variable you throw a dice. The second question, you need to throw 2 dices and they need to both be in a certain range. Of course you're gonna get different result... the flaw is thinking it's the same question
Each question only has one variable, length or area. Since the shape made is always a square then from the area we can calculate the lengths of both edges. Therefore it’s exactly the same question since each length has a specific area associated with it and vice versa.
I think Bertrand's Paradox and considerations like the Grue/Bleen problem show that in probability we do not normally KNOW where to start, but that there is a psych bias built into a choice of starting places, a bias which a Bayes analysis may show to be perfectly natural and of pragmatic value, but lacking in epistemic value. If you think accuracy of descriptions is not a problem, consider the joke: "There's a fine line between fishing and merely standing on the pier dangling your line in the water." - two situations empirically indiscernible to the observer on the pier. As you might say: "He is FISHING if trying to catch fish, but DISHING if merely dangling a baitless line."
The solution here is actually quite straight forward. At the start of the video, we are told that the factory cuts squares such that the edges are between 1 ft and 3 ft. This information determines that our wood squares will be cut based on edge length, not based on area. With no further initial information, we must assign a uniform probability distribution to the range of possible lengths. Up to this point, the video is correct. Since length maps to area through an exponential function, and not a linear one, our uniform distribution is lost. The area range (1 sqft - 4 sqft) that corresponds to the length range of 1 ft - 2 ft does account for 1/2 of the probability, though it only accounts for 3/8 of the total areas possible. You are, in fact, building the probability distribution for the area, as you map the length to the area.
"With no further initial information, we must assign a uniform probability distribution to the range of possible lengths." Wrong. We must admit that, unless we're told the probability distribution of lengths (or of area, it doesn't really matter which we are given), we can't answer the question.
@@brycerosenwald2915 //Abstract The principle of indifference (PI) states that in the absence of any relevant evidence, a rational agent will distribute their credence (or 'degrees of belief') equally amongst all the possible outcomes under consideration.//. So why can't he assumes that the length has uniform distribution without being told by the question?
@@iwatchtvwithportal5367 well, I think you answered your own question - because it's an assumption. Assumptions are sometimes warranted, but when a question doesn't include enough information, or lends itself to multiple interpretations, you can't just make an assumption and say, "yeah, that's the answer to the original question!"
@@iwatchtvwithportal5367 I think maybe I misunderstood this comment. There are cases where PI works, but this paradox is specifically meant to be a case where PI breaks down. Specifically, the possibility space in this question (i.e. the range of square sizes) can be described in multiple ways. When you describe the space one way - in terms of side length - and apply PI, you get one probability distribution, whereas when you describe the space a different way - in terms of area - and apply PI, you get a different probability distribution. But you shouldn't be able to derive two different probability distributions from the same information - so what's going on?
Perhaps the focus here is in the wrong place. The illicit move was obvious as it occurred, so much so that the explanation bordered on _over_-explanation. But what wasn't obvious was how the principle of indifference plays an important and irreplaceable role in science. How is it that we're supposedly stuck using it in anything but forced toy puzzles?
The probability of area is NOT 3/8. It's 1/2 just like the probability of length. Here's the proof: This is a basic change of variables probability problem dealing with probability density functions (pdfs). Let x represent the length of the square produced, where x is a uniform random variable from 1 to 3. Then the pdf of x is fx(x)= 1/(b-a) when x is between a=1 and b=3, and fx(x) = 0 when x is outside of a and b. So the probability that x is between 1 and 3 is the integral of fx(x) from 1 to 3 which equals 1/2. Great. Now here's the part where he messed up. Let Y = X^2 represent the area of the square produced. In order to get the pdf of Y, we use the formula fy(y)=fx(x)*(dx/dy). Thus the pdf of y fy(y)=1/(b-a)*[(1/2)*(1/sqrt(y))] when y is between a=1 and b=9 and fy(y)=0 when y is outside of a and b. So the probability the area is between 1 and 4 is the integral of fy(y) from 1 to 4. Which equals 1/2. He assumed that the pdf of y must also be fy(y)=1/(b-a) which would give 3/8.
"Let x represent the length of the square produced, where x is a uniform random variable from 1 to 3." No one ever promised that x would be a uniform random variable. You've just invented that bit yourself. "He assumed that the pdf of y must also be fy(y)=1/(b-a) which would give 3/8." You made the exact same assumption for x. Why is it wrong when he does it for y, but right when you do it for x?
Ansatz66 Good question and good point. You're right. I can't know the true distribution without having more information. The point of my post isn't to claim that the distribution is in fact uniform, the point is that you can't assume the same distribution when doing a change of variables. I just arbitrarily chose uniform for this explanation because it's an easy example and gives 1/2 for the first step just like in the video, but a different answer than 1/2 after the second step to show where the math might go wrong in the video. It could in fact be any distribution. However proving the example with a more complicated equation would be a lot more work to prove
"..where x is a uniform random variable from 1 to 3. " , " So the probability that x is between 1 and 3 is the integral of fx(x) from 1 to 3 which equals 1/2. Great." if x has equal probability of being anything between 1 to 3, then x has %100 probability of being in between 1 and 3.
Why do youtube comments always think they solved old paradoxes... Every. Single. Time. No. Exceptions. Maybe the youtube comment section is both the dumbest and the smartest place in earth... IT'S A PARADOX
We RU-vid commentators are usually asses. That's given. But this video is itself flawed in a typical way: it badly misjudges what its audience will take to be obvious or commonsensical. It spends its time explaining how going by length will give different results from going by area, even though those two methods giving different results is exactly what you'd expect. What actually needs detailed explaining is why anyone would expect them to give the same results. And if that missing half of the conflict doesn't show up to defend itself there just is no conflict. No paradox.
1) This "Bertrand things" is no paradox, so there is nothing to solve here. 2) Maybe there are plenty of people commenting, dumb and smart. Those with most widely accepted answers become more relevant, and eventually get to the top.
I think you used a bad example. (Ignoring the decimals), there is no possibility of getting an area different from 1, 4 or 9. So, this reduces the probability problem to the same as the length problem.
There is no paradox here, and the problem isn't in what is said but in what is not said. What is not said is the distribution. If the distribution of edge length is flat then the probability of 1 to 4 feet is still1/2, if the distribution is size probabilities is flat for area thing the probability is 4/9ths (not 3/8th)
If we should assume that the sick horse will perform worse, reasonable but not certain given general knowledge on sickness, horses, performance and races, we should also assume things about factories and their products. A uniform distribution is not very probable. If I had to guess, I'd go for tight normal distributions around a few assumed standard lengths such as 1, 2 and 3 length units. If I'm not supposed to do that, then why am I supposed to assume things about the poor horse?
I just feel like the propability of one case depends on the other...if you assume that lenghts are distribued equally, then it is simply more likely for the areas to fall somewhere between 4 and 9...those are not equivalent statements...
Yes, but if you assume the areas are distributed uniformly then the statement about the lengths is not equivalent either! The point is that you can't apply the principle of indifference to continuous variables without weird things happening.
The cuts are from 1ft (1sqft area) to 3ft (9sqft area). Meaning that the average cut will have 2ft (4sqft area). The average cut will be 2/3rds of the max size. The average area of 4sqtft will be 4/9ths of the max possible area of 9sqft. (2/3)^2 = 4/9. This part was not a paradox, but a fallacy. Just draw 9 squares in a sheet. Assume the first square is a given as the machine will not cut less than that and go from there.
This "paradox" assumes a uniform distribution of side lengths, then also assumes a uniform distribution of areas. The point is, if side length is uniformly distributed, area won't be. Think about this: If you roll 2 dice and take their sum, the sample space has 11 elements: {2, 3, 4, 5, 6, 7, 8, 9, 10 , 11, 12} but you wouldn't say that "7" has a 1 in 11 chance of occurring. That's because the distribution isn't uniform.
The two questions/problem are not the same, so I don't like how they are constantly presented as the same problem in this video. The language chosen simply chose between two different problems and formulas. Two different problems result in two different formulas. But he keeps saying they are the same - they do not solve the same problem. This film has the wrong focus - rather than a paradox (which there are better examples of) this teacher should be promoting how to carefully and accurately determine exactly what problem you seek to solve, that problem definition is key and that use of language in the proper formulation of mathematical solutions is extremely important.
Am I the only one who is getting mind fucked here? when you measure Length you start from Zero, not one. I know you were trying to bring a new analogy to understand this, but it makes it more confusing. also those are not "edges" they are called Sides. but anyway I liked the fact that he is From UofT :)
This is not a paradox at all. Assuming the probability of getting a square size 1-5 is exactly as big as the probability of getting a square size 5-9 is just plain wrong. Im actually a bit disappointed you want to sell this as a "paradox".
Using a factory as an example is not that great because the answer will always beca >90% probability that each subsequent square of wood would be exactly the same, at least for a limited run. The reason is that the cost and effort to reset the milling machines for just one piece of wood would be so costly that if would almost certainly never be done. Using a factory as an example eliminates the randomness that you really need in an example of this type.
Without diving in too deep into comments amd whatnot, this presentation is overcomplicated. There are simpler ways to explain Bertrand's box. The problem is that an self-reassured person will find any justification that 1/2 is correct. Introducing all this math only makes the problem less intuitive for the layman, which is no help to the layman; He doesn't want to deal with all this mathematical mumbo jumbo. Bertrand's box is a profound problem because it's not just an unintuitive problem-it's a test of open-mindedness. If you try explaining it to someone who's self-reassured, they will "reject your reality and substitute their own". Explaining it in more complicated terms doesn't help such a person "see the light". This video only caters to people who already know the solution whom are looking for various proofs.
This feels like a false paradox (the one specifically presented, at least). The machine doesn't select an area, it selects a singular value and the area is a mere consequence of it. When you consider area, 4 square whatevers, it could be 2x2 or it could also be 1.5 * 2.666.... , or any infinite number of multiplications that give you 4. But if an area of for 4 necessarily means exclusively 2x2 (i.e. we exclude all rectangles and only consider the squares), the "paradox" here is that the space being used just isn't the actual space of the problem you're dealing with. i.e. the problem is bad modelling, not statistical inconsistency, at least not on this particular example.
The problem here is that the principle of indifference is simply wrong. It only applies if each possibility is equally likely. In the case of the square problem, only one variable, the side lengths, or the areas, can be distributed uniformly. Because each variable's value depends on the other one's value, they cannot both be distributed uniformly. The reason two different solutions were found is that the problem was ambiguous. The first solution is correct if there is an equal probability of the squares having any side length, and the second is correct if the squares have equal probability of having any area.
Joseph Noonan The principle of indifference applies _only_ if you don't know how likely something is. It tells you to treat all possibilities as equally likely. Which seems to be very rational. But there seems to be no unique way to divide a problem into different possibilities, which then leads to inconsistent probability distributions recommended by the principle of indifference. So how can the principle of indifference be rational? That's the paradox!
Mathematical principles work because of assumptions made for a particular system. Comparing two results from two sets of different assumptions is not a paradox! It is an error in human judgment, not an error in the principles of mathematics and science. Science and Math will remain flawless, even if the human interpretation of them falters. If you go back and define your assumptions correctly and explicitly as per the requirements of the principles you apply, you will come to a different conclusion. And so will this fallacious paradox vanish.
you are taking the probability of a PIECE of wood been from the area of x, not the total of cubes, the line of the square roots should be in the same SIZE of each other, changing just the position of the 1-2-3 for 1-4-9. The cubes still go out in the same way, but if you want to know WHAT IS THE ODD OF ~ONE~ PIECE OF WOOD come out been with 1-2-3, now you have 3/8.
Isn't this like, totally bad math? When you ask how many squares would be no longer than 2 ft, you have to take into account both sides of the square. So that would be 1/2 x 1/2 = 1/4. And when you ask, how many squares would have area no bigger than 4 square feet, that is a totally different question, and range. No wonder the answer is different, because you also count squares 3ft by 1ft = 3sq feet, where these squares are not counted in the first question. There is no paradox here, just some professor without understanding of basic kindergarten level math.
@@joshuaronisjr yes. And it is also counted in the first question, as sqrt3=1.73. And this is less than 2. So the board sqrt3xsqrt3 is counted in both questions but board 3x1 is counted only in second question.
another bullshit video from this channel. the probabilities for the same case of cutting can‘t be 1/2 and 3/8. the shorter part of the m2 line has the same probability of 1/2 as the second longer part of the line. because it is determined by the probabilities of the lenghts.
