joigus

Unfortunately, no. Murphy's law is not an actual law of probability, but a humorous observation on the nature of our expectations.

I don't think @MigL will like the way this is turning...

Well, the word 'random' is a tricky one. As I said, there is no unique way to define 'random'. People tend to think of random as synonimous of 'unbiased'. Bertrand's paradox --which I also mentioned before-- shows that the premise of randomness as one of total unbiasedness (equal probabilities for all the values of a variable within a range) gives different probabilities for different variables that equally describe the same problem. Namely: Consider an equilateral triangle that is inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle? Then Bertrand proceeds to calculate this probability by different methods, all equally unbiased, but with respect to different variables (two points chosen at random, a point and an angle, etc). The answer is different depending on which variables you choose. The illusion that 'random' means something precise comes from: 1) Choosing a discrete set of outcomes, and 2) Assuming there is a probability distribution that's 'written in stone', like, eg, Laplace's rule of equal probabilities, some principle of symmetry infering that (example, the fair coin), etc, so that assigning probabilities is reduced to a counting problem. Otherwise, we need some kind of law or fundamental principle that tells us what the distribution is, like we have in statistical physics, for example.

Big arthropods were peculiar (if not exclusive) to the Carboniferous period, due to the high oxygen levels, rather than gravity. Number of appendages seems to be more of a developmental accident, as most members suggest. Embryonic development tends to follow body plans long past initiated.

Yes, provided you are invited to just pick a letter. But even in that case, if you asked people to try to guess an answer that makes it correct, the sheer fact that people wanted to get it right would shift the probability distribution to more than .25 chance, only because they think that's the right answer! I think that's what @studiot meant when he compared this to the Monty Hall problem. I don't think it's a pure Monty Hall problem with conditional probability hidden in some semantic way. I think there's also a more-or-less hidden self-reference here that contributes to the paradox. Yes, but then you're assuming a probability distribution: .25, .25, .25, .25 and you cannot escape the paradox either. .25+.25 = .5 so .25 would no longer be 'correct'. It is designed in such a way that, if you want to get it right, you'll get it wrong.

The way I see it, the most offending term in the posing of the problem is the mischievous use of the expression 'at random'. Whether we adopt a frequentist approach or an aprioristic one, distributions such us, eg, A: 31.7 % B: 19.2 % C: 16.1 % D: 33 % and, A: 12.5 % B: 23.9 % C: 11.2 % D: 52.4 % are exactly as 'at random' as each other. Too often, people confuse 'perfectly random behaviour' with 'equal probabilities' (Laplace). The best cautionary tale I know about this is Bertrand's paradox: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

Dear Studiot, it would be shocking indeed that I recognised a character from a novel I haven't read! I love British culture, but I cannot compete with you in that respect. I'm looking up Wyndham as we speak. Mind you, I must keep an eye on domestic affairs, literary and otherwise.

I'm afraid I haven't, but coming from you, I'll look it up. Thank you.

Well, that's why @TheVat 's comment comes in handy too. If 'at random' means something like 'throw a die' = 'just pick one' (Laplacian probability), it would be 1/4 each. If it means 'take a guess based on intuition' there's another probability distribution. If it means 'give the problem to machine X' there's another, and if given to machine Y, still another. If it's: 'ask many people and calculate the probability á la Pearson', then probably (sorry for the pun) it would be zero, as a continuous distribution in (0,1) assigns zero probability to any particular value (and, as you know, zero probability doesn't imply impossibility). Etc. I think the paradox is a well-thought-out one and it illustrates a problem of using the concept of probability in too-loose a way.

I think the essence of both these comments is pretty similar. Yes, knowledge, or even ballparking, intention, etc, of the person guessing essentially changes the probability distribution. I have developed the habit to actually ask, 'what do you mean "random"? According to what probability distribution?' Most people get confused, but I think I know what I'm asking. The moment you know something, or venture to guess something, or think you know something, the probability distribution of your answers already changes.

Son: "Hey, dad. I have an imaginary girlfriend" Dad: "You can do better" Son: "Thanks, dad" Dad: "I wasn't talking to you"

It has a flavour of it. Yes. I think the paradox is always inherent in the question, by the way. You just missed the last step. In order for the probability to be 50%, 25% must be wrong. Therefore the probability must be 0%. But don't forget the answers must be chosen at random, so no answer can really be right, but 25%, but then it's 50%, but then... An so on. Similar to Epimenides' paradox, yes. It has a looping structure. This one is easy, and not paradoxical, AFAICT. The right answers are a) and d) Mathematicians do not suffer paradoxes, I think. They know the problem must be ill-posed in the first place, and so set out to identify the illness. Engineers, OTOH, are too practical. It's more for the likes of physicists to break a sweat on these things. By the way, the words 'at random' are a potential minefield, as the don't mean anything in and of themselves. If this discussion proves to be any popular, we might end up discussing that.

If by that you mean paradoxes exist only in our reasoning minds, and the world of facts always get past them, I think you're probably right.

In my (reasoned) opinion, only the aprioristic concept of probability produces a paradox here. Experiments cannot be paradoxical. Therefore only one answer should be correct from an a posteriori POW, ie, actually conducting the experiment and checking the right box. What's your take on this?

Not true. The square roots of positive numbers, yes. They are two-valued, with one value positive, and the other, negative. For complex numbers, it's more involved. For negative numbers, you need complex numbers already.

So in what way is this different from GR, except for a re-wording of the usual concepts? Can you offer any new insights provided by this re-wording? I mean, you can define sets of coordinate vector fields, and introduce a smooth metric on your manifold. Then obviously the points where g(dx,dx)=0 are the boundary separating those for which g(dx,dx)>0 from those for which g(dx,dx)<0 This is tautological.

One such model would go against relativity on a number of levels. For starters, there's no "present" in relativity. Not even in special relativity. In the De Sitter model, eg, every observer carries its past, present, and future. Observer-dependent are the key words.

Not the same, criticising prosthetics for your tongue and lips (writing) than prosthetics for your brain (AI). I've seen AI work like a dream in fields where there is general consensus on principles, definitions, limits of applicability, etc (example: financial maths, where everything consists of arbitrary definitions), and fail miserably where there are important divergences of opinion/interpretation, bounds unclear, etc (example: cosmology, ultimate laws of physics, like, eg, the local gauge principle, or realism vs determinism, etc). It's very clear to me that in those areas AI tends to either shamefully hedging the bet or plain getting it wrong (missing the necessary nuances altogether). OTOH, whereas maths leans heavily on rigour, physics does not. Physics leans heavily on experimental fitness rather. Mild lack of rigour (or serious, as the case might be) is forgiven if experiments are well accounted for within allowed ranges of parameters and variables. Sorry. This is not my debate, but I had to jump on this, because I saw a logical flaw about an argument that has occupied my mind lately.

It is. You could in principle trace your family tree back to the Middle Ages in Spain. In actuality, of course, very few people --if any-- can do that. Trusting that your great-great-great grandparents and beyond had everything under control when it came to procreating is another matter. 😅 Some Basque family names, like Urrutikoetxea ("house from afar") or Etxebarria "new house" are a bit more dicey as to their origins being based on a gild, or a piece of land.

@HbWhi5F , could you rephrase the question, please? Obviously, said function cannot be bigger than its value at x=1. It cannot be less than zero either, so there you go.

Why doesn't the spoiler work? It's the plural of @Genady 's visual answer.

I think we tend to overestimate the immediate evolutionary value of high-level reasoning and data gathering that we see today in sophisticated science, at least for the individual and the close kin. I doubt individuals or clans tens, hundreds or even millions of years ago did anything much in the way of, eg, considering counter-arguments to a given argument. The most important thing was probably to act quickly in a way that's not obviously contrary to survival conditions, and has worked within reasonable limits. And now that I think of it, @exchemist has very much answered to this in a way I would subscribe to. So sorry I initially misunderstood the premises.

I still would like to know whether @Otto Kretschmer means 'motivated reasoning' in a sense that specifically can be phrased as, As defined in https://www.sciencedirect.com/topics/psychology/motivated-reasoning

Thank you. I hope people who know more about this than you or me can illuminate the nuances. I must say I tend to see much that's going on among humans (in evolutionary terms) as a product of the needs of small clans having to struggle for survival 50'000 years from now until the present. Agreed. Cephalopods and corvids come to mind too for relatively complex congnitive features. Sea mammals and great apes immediately spring to mind, of course. I also find difficult to believe that dinausaurs didn't develop fairly complex cognitive abilites, given that they we here for more than 200 million years.

Where do you place the starting gun, then? I'm thinking now that I may have misunderstood what the words 'motivated reasoning' mean in psychology. If I understand correctly, it's about emotional bias. From https://www.sciencedirect.com/topics/psychology/motivated-reasoning: