Yes, e is between 2 and 3 and Pi is between 3 and 4. There are geometrical lengths corresponding to each number.
>Isn’t it definitionally impossible to pick it as a “point along the line”?
No, it's mathematically possible to have a random process which picks a random real between 0 and n, with equal probability. Imagine it akin to throwing a dart at a line and picking the point it lands on as the number.
Since there are only countably many rationals and uncountably many irrationals (i.e. not just infinitely more, but so many that you could never pair off the rationals with the irrationals, there are just too many) on any such length of the real line, chances are the number you end up with is overwhelmingly likely to be irrational.
And it’s not “overwhelmingly likely” as in there’s a 99% chance or whatever. If you choose a random point on the line, the probability of choosing a rational is zero.
Yep, exactly. I glossed over that detail a bit because explaining how a meagre set has a truly zero probability of being picked, while technically still being a possible result of a random process, is a bit messy to wrap your head around colloquially.
> If a thing is in my pocket, there's an above zero probability of me picking it when I randomly take a thing out of my pocket.
This is only true if there are only a finite number of things in your pocket, though… I think an analogy is how we always have 1/n>0 for any finite (positive) number n—and yet, 1/infinity=0.
> Do math people not feel the need to explain themselves when they state things that defy common sense everyone except math people agree upon? Is that part of thinking you're 'smart'?
It's a pretty basic thing covered in undergrad prob/stats classes. We don't re-explain it every time we use it for the same reason computer scientists don't re-explain the halting problem every time it comes up.
I’m struggling to understand how a thrown dart could land on an irrational number. It seems definitionally that any physically realized outcome must pertain to a rational number because it is impossible to physically measure one at any level of precision.
It is possible to write a random process that returns 5 or pi with 50/50 odds so this isn’t a very compelling argument that it’s possible. I don’t feel the semantics of picking a random point along a number line is gg solved just by appealing to the existence of uncountably infinite irrationals.
By most people’s definitions of random points along the number line, including the dart throw, it seems to me the probability of getting an irrational is 0.
Invoking the number of possible outcomes has bad feeling implications. For example if your set is 1 2 3 pi 4, then the probability of getting an outcome in [3,4) is higher than [2,3) and that seems like it’s breaking the intuition of what the line represents. Like as a stupid example say we only include the irrational numbers between 9 and 10 and pick a random point between 1 and 10. If the random method uniformly sampled a point along the line by distance we would suggest a 90% chance of getting a rational number <= 9 and a 10% chance of getting an irrational number above 9.
But if we sample by naive odds you’d probably claim there’s a near 100% chance of getting an irrational number above 9 because there’s an uncountable infinity up there.
> It seems definitionally that any physically realized outcome must pertain to a rational number because it is impossible to physically measure one at any level of precision.
Sure, that's correct, but it isn't what people are talking about here.
> By most people’s definitions of random points along the number line, including the dart throw, it seems to me the probability of getting an irrational is 0.
That depends on the number line you're using. You can say that irrationals don't exist and you won't lose anything. But if your number line includes the reals, then the rationals form 0% of it.
> Invoking the number of possible outcomes has bad feeling implications.
That isn't how this is measured. You don't want to compare a count to an area. For probability, you need to compare like with like. A number line is one-dimensional, so we consider one-dimensional areas, or "lengths".
The interval from 0 to 50 has length 50. How much of that length is occupied by rationals, and how much by irrationals?
Each value is a point with no length. So, to measure the rationals, we assign to each rational point an interval that contains it. We will estimate the total length occupied by the rational numbers within the interval as being no greater than the total length of the intervals we put around each one.
Since there are only countably many rationals, we can use an infinite series with a finite sum to restrict our total-length-of-intervals to a finite amount. (Rational number one gets an interval 3 units wide. Rational number two gets one 0.3 units wide. Number three gets one 0.03 units wide. What do all these intervals add up to? Four thirds.) We can scale those intervals however we like. We will scale them down. If our first set of intervals had total length 20, we can multiply them all by 1/400 and now they'll have total length 1/20. The limit of this process is a total length of zero, which is our upper bound on how much of the length of our interval is occupied by rational numbers.
Since zero is also a lower bound on any length, we know that the total length of the interval occupied by rational numbers is exactly equal to 0. It is then easy to calculate the probability that a randomly chosen value from this interval will be rational: it is 0 (the amount of length occupied by rationals) over 50 (the total amount of length).
> Like as a stupid example say we only include the irrational numbers between 9 and 10 and pick a random point between 1 and 10. If the random method uniformly sampled a point along the line by distance we would suggest a 90% chance of getting a rational number <= 9 and a 10% chance of getting an irrational number above 9.
This seems to be just you being confused over the concept of a uniform distribution.
> This seems to be just you being confused over the concept of a uniform distribution.
Try and follow the example again.
The distribution is all rationals 1-9 and all numbers 9-10.
Sampling uniformly such that each distance is equally likely across the line gives at least a 90% chance of choosing a rational.
Sampling uniform by elements of the set gives a 0% chance of choosing a rational.
The problem with the latter is that even though you’re claiming to be randomly sampling the _line_ you are never going to sample the first 90% of the line length because you are instead sampling the _distribution of set elements_.
You are NOT more likely to throw a dart that lands in 9+ just because you have magically introduced an infinitely tense series of irrationals in that range.
> The problem with the latter is that even though you’re claiming to be randomly sampling the _line_ you are never going to sample the first 90% of the line length because you are instead sampling the _distribution of set elements_.
This is all in your head. Who are you responding to? Where did your three claims ("sampling uniformly by distance from 0" / "sampling uniformly by element count" / "randomly sampling the line") come from? What does "sampling uniformly by distance" mean? Uniform sampling is done by count for discrete sets and by area for continua. You have yet to mention a discrete set.
It is the difference between picking a random point along a line and picking a random number from a set. A dart throw will not land in the range of [9,10) more often than [1,9) simply because we are considering irrationals in the former.
These are both uniform. But the outcome is different
You're never going to get anywhere without defining your terms.
As you originally pointed out, a physical dart can't hit a single point on a number line. It will hit an infinite number of them simultaneously. This is true whether you're worrying about rationals or reals.
But if you have a dart so sharp that its tip is zero-dimensional, one that can hit a single point on a real line, and you throw it at a composite of the rationals from [0,9] and the reals from [9,10], it will have a 10% chance of hitting an irrational number (within [9,10]), and it will have a 90% chance of missing the line entirely, striking one of the holes in the rational interval [0,9]. The chance of hitting a rational number will not improve from 0.
Do you have a model of uniform selection in mind, or do you find that it's easier to say the words without assigning them any particular meaning?
> Sampling uniformly such that each distance is equally likely across the line gives at least a 90% chance of choosing a rational.
Let's say the numbers are targets on the line. Your distribution implies the range 1-9 is less dense with targets than the range 9-10. Doesn't that mean you're less than 90% likely to hit something between 1-9?
> You are NOT more likely to throw a dart that lands in 9+ just because you have magically introduced an infinitely tense series of irrationals in that range.
If we turn this around, by forbidding a bunch of values in the 1-9 range from being hit, then won't the probabilities get skewed towards the 9-10 range?
No, because a dart throw is not a uniform draw from set elements. Is a uniform draw of length which the inclusion of irrational numbers does not affect. You are 90% likely to throw something in the first 90% of the line. It doesn’t matter if we say we will round anything in [1,2) to 1. There’s a ten percent chance of falling in that range.
Not a 0% chance because there happens to be an uncountable infinity number of options in [9,10)
We are so, so far beyond that point already. The complexity of the world economy is beyond any one mind to fully comprehend. The microcosm of building black-box LLMs that perform feats we don't understand is yet another instance of us building systems which may forever be beyond human understanding.
How is any human meant to understand a billion lines of code in a single codebase? How is any human meant to understand a world where there are potentially trillions of lines of code operating?
When your house is on fire and someone says "get out", certainly grabbing a jerrycan of gasoline and dousing yourself in fuel is worst than just getting out?
This is the beginning of the end of the public internet, imo. Websites that aren't able to manage the bandwidth consumption of AI scrapers and the endless spam that will take over from LLMs writing comments on forums are going to go under. The only things left after AI has its way will be walled gardens with whitelisted entrants or communities on large websites like Facebook. Niche, public sites are going to become unsustainable.
Yeah. Our research group has a wiki with (among other stuff) a list of open, completed, and ongoing bachelor's/master's theses. Until recently, the list was openly available. But AI bots caused significant load by crawling each page hundreds of times, following all links to tags (which are implemented as dynamic searches), prior revisions, etc. Since a few weeks, the pages are only available to authenticated users.
Now account for the amount of mental overhead required for the average person to calculate change or coinage of a random amount in base two coins, as opposed to multiples of 5 or 10, and see if your 3.19 coins per transaction really saves you time.
For an interesting exploration of how even the simple physics of two blocks hitting one another can lead to surprising conclusions, I'd highly recommend this 3Blue1Brown video called "Why do colliding blocks compute pi?": https://www.youtube.com/watch?v=jsYwFizhncE
A lot of times with math I don't think the word "why" should be used. Those blocks computing pi was a good example where they "proved that they compute pi", but not really why. What is the real distinction I'm trying to make here and how to explain it?
It's been pretty incredible watching these companies siphon up everything under the sun under the guise of "training data" with impunity. These same companies will then turn around and sic their AIs on places like Youtube and send out copyright strikes via a completely automated system with loads of false-positives.
How is it acceptable to allow these companies to steal all of this copyrighted data and then turn around and use it to enforce their copyrights in the most heavy-handed manner? The irony is unbelievable.
It's always very impressive to me seeing our ability to detect such obscure objects in advance getting better and better. We'll soon have such good detection capabilities that we may start to take these kinds of predictions for granted the same way we take accurate weather forecasts for granted. Can't wait to see the local meteorologist talking about actual meteors.
I've had this thought about payment cards for a long time now. Corporations are pulling a reverse Superman III on us and keeping our pennies and dollars on these gift cards and payment cards that can never quite be zeroed out. If you added up all the lost change from all of these cards over someone's life it could be a fairly non-trivial amount of money. Add this up over a whole economy of people and you have millions of dollars of change that you've siphoned out of people by making it too difficult to spend.
There should be a law mandating the ability to convert gift cards or payment cards back into cash, or to reverse the transaction onto a credit card.
Ready to cry a little bit? "47% of American adults have unused gift cards, and the total value of these unspent funds in the U.S. is around $23 billion."
Are these cards actually paid in advance? Or is much of their bulk effectively debt that would need to be repaid with a pretty low probability? See various coupons, mail-in rebates, points that you can only redeem in a particular store, etc. They look like a great deal, and are cheap to issue because only some of them are going to be redeemed.
While it's not explicitly stated, the stats indicate this is gift cards that have been paid in advance.
That means "Get $20 back when you spend $100" and "Get a $20 gift card when you spend $100" wouldn't be included. Not sure about "here's $20 to spend on your next visit"—I suspect that's really the same as the above, financially, just presented differently, in that there's not $20 sitting in 200,000 individual accounts waiting to be spent.
I worked at a job that had a cafeteria where you have to put money on a debit card, then pay with the card. It's a penny stealing operation for sure since a lot of food items are charged by weight so they are odd numbers. I worked at that location for about a year and on my very last day, my balance hit $0.00 by almost pure luck. I have never been more satisfied.
In California, at least, it's quite strict. If it's not used for a while and the balance is over $10 it goes through escheatment and ends up with the state. But if it's not expiring you still have to honour the card. Then get the money back from the government.
At least in California, any gift cards under $10 are redeemable for cash by law. But thats not saying the merchant will make it easy to do so. I usually have to ask for a manager and wait for a bit.
And increasingly, the minimum gift card amount is going up. You used to be able to buy Switch eShop gift cards for $5, $10, $20, and up. Now the minimum amount you can buy is $20:
Very frustrating for me, given that I still use these regularly for all my e-shop purchases, and can sometimes come up a single dollar short of what's necessary to complete the transaction.
Sony still lets you go down to $10 for PSN cards, but I'm skeptical that they'll keep it around much longer.
> Corporations are pulling a reverse Superman III on us and keeping our pennies and dollars on these gift cards and payment cards that can never quite be zeroed out.
Use the low-balance card to pay for part of a larger purchase.
Yes, e is between 2 and 3 and Pi is between 3 and 4. There are geometrical lengths corresponding to each number.
>Isn’t it definitionally impossible to pick it as a “point along the line”?
No, it's mathematically possible to have a random process which picks a random real between 0 and n, with equal probability. Imagine it akin to throwing a dart at a line and picking the point it lands on as the number. Since there are only countably many rationals and uncountably many irrationals (i.e. not just infinitely more, but so many that you could never pair off the rationals with the irrationals, there are just too many) on any such length of the real line, chances are the number you end up with is overwhelmingly likely to be irrational.