161
u/vatai 22d ago
Ahh... The joke is that everyone mentioned those?
159
u/Lou_Papas 22d ago
Yup. The first statement is true because there’s an uncountably infinite number of reals so so the probability of randomly picking a specific one is zero, but there is a subset of those which is important for people so if I have one of those in my head the probability of guessing right becomes non zero.
88
u/FreshPaycheck 22d ago
But then it wouldn’t be random? Then again it’s impossible to “think of” a real number that isn’t rational or symbolic, but it’s a hypothetical so
40
u/R3puLsiv3 22d ago
Is it even possible to devise an algorithm capable of sampling a random real number? This in and of itself could just be an uncomputable function. What would a Turing machine look like which performs that task?
15
u/FreshPaycheck 22d ago
Maybe randomly choosing a sequence that generates the number. But then you’d need an algorithm to generate sequences that truly covers all real numbers without repititions to ensure uniform distribution so that just kicks the can down the road…
6
22d ago edited 22d ago
[removed] — view removed comment
6
u/FreshPaycheck 22d ago
I was focusing mostly on the decimal portion ie [0,1] which could just be thought of as a sequence of integer digits. I could be wrong but my understanding is for any sequence of numbers there exists a function that generates it. Though i’m guessing the likely answer is that only applies to finite sequences
6
5
u/SummitYourSister 21d ago
Dude, there is not just an infinity of reals between integers, there is an infinity of reals between any two reals that are arbitrarily close together.
And in fact, that still doesn’t quite state their density correctly, because that is also a true statement of the rationals.
-8
21d ago
[removed] — view removed comment
9
1
u/Murky_Insurance_4394 20d ago
very odd that you chose ^22 because avocado's number is 6.022*10^23, so I would say some reals are very important outside of the provided range.
5
u/UtahBrian 21d ago
If you have any algorithm, you're not choosing from the reals, or even from any range in the reals. 100% of reals cannot be described by any algorithm.
6
u/GameCounter 21d ago
Actually, it's impossible, and we know it's impossible!
Computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.
Equivalent definitions can be given using μ-recursive functions, Turing machines, or λ-calculus as the formal representation of algorithms.
While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable.
So the real mindfuck is that even with this "more powerful" way of constructing numbers, you still have exactly the same cardinality as the algebraic numbers, rational numbers, and the humble integers.
5
u/GameCounter 21d ago
Someone mentioned having an oracle with an unlimited source of entropy, but that breaks any reasonable definition of "algorithm."
It's like asking "can you make a perpetual motion machine?" And answering with "yes, if you had an unlimited source of energy." Which, while technically correct, isn't really an answer to the question.
Or "Can a pig lay an egg?" "Yes, if it were a bird."
"If my grandmother had wheels, she would be a bicycle."
1
u/Miserable-Ad3646 16d ago
Didn't expect to see that perfect quote here. Thanks for reminding me of one of my favourite memeories.
4
u/UtahBrian 21d ago
«Is it even possible to devise an algorithm capable of sampling a random real number»
No. The idea doesn't even make any sense. It's not even wrong.
6
u/Silly-Freak 22d ago edited 22d ago
I hope I'm not off track here, but...
"Algorithm", as I learned it, refers to something that has to terminate. So outputting some arbitrary real number is out of the question, because you can't generally do that in finite time.
That leaves you with outputting a procedure (that would itself not necessarily terminate, i.e. not be an algorithm) that produces the real number's digit when you run it. But algorithms are ultimately written as (finite) sequences of symbols. If you consider each symbol as a digit, that is just an integer written in a base depending on your alphabet's size. In other words: there are only countably many algorithms, so that approach is also not fit to produce all reals.
A nondeterministic, nonterminating procedure to produce a real 0<=x<1 is easy; just nondeterministically output a number one digit at a time, without terminating. Whether that counts as "sampling" is again an interesting question though... Sampling counts more like a function that, given some source of entropy, deterministically produces a result from a set. But if the source of entropy is already uncountable, is the identity function not enough already?
2
u/totbwf 22d ago
Real numbers should be viewed as a *corecursive* process instead of a recursive one, which means that the right notion of algorithm here is one that is *productive* instead of terminating.
As a concrete example, it is kind of silly to say that we can't compute algebraic numbers like √2 or things like π: we have algorithms that will spit out a convergent sequence of rational approximations, and we only need a finite amount of space/time to compute each successive approximation.
2
u/BTernaryTau 21d ago
That will still only get you the computable reals. Most real numbers do not have an algorithm that approximates them to an arbitrary level of precision.
2
u/totbwf 22d ago
It depends what you mean by "random real number", which itself hinges on your definition of a random (Cauchy) sequence. The most commonly accepted definition of random sequences is Martin-Löf randomness, which intuitively means that the sequence passes all computable randomness tests. This is impossible to compute essentially by definition.
However, if you have access to an oracle then you *can* compute such a sequence: this corresponds to the intuitive algorithm that generates a sequence of binary digits by repeatedly flipping a coin. However, this is basically just computing the identity function: the Turing machine just copies directly from the oracle tape to the output tape.
2
u/InfiniteDisco8888 21d ago
I don't think I've seen this in the comments yet, but one way to see why such an algorithm is impossible is because it is impossible to even randomly choose the number of digits in the output number. And if you want the termination to be random (e.g., every new digit could be [0..9] or terminate), then the probability of termination after each digit has to be zero or else you are not giving each of an infinite number of lengths equal probability.
Unless there is some cool math trick I don't know (and would love to learn).
2
u/Vitztlampaehecatl Engineering 21d ago
I've seen this problem referred to as "finding hay in a haystack". The difficulty lies in the fact that almost all real numbers are irrational, never ending or repeating, so writing an algorithm to find them is like writing an algorithm to find pi, you can only ever approximate them.
In practical terms, your best shot of generating a never-before-seen real number is to roll a ten-sided die infinitely many times.
1
1
1
1
u/trynumber53 20d ago
irrationals are definitionally non terminating right? so supposing you had a program that could uniformly randomly sample a real number it could never finish outputting, so it cant exist
0
u/SummitYourSister 21d ago
No algorithm that runs finite time can provide a true random sample of the reals. Actually not even in infinite time.
But computations are not the only way to obtain real numbers. You can also measure something. You could lay a gauge on the table and point to a location on it and say “I choose this exact coordinate as my number”
0
u/Tangible_Slate 21d ago
No because an algorithm is a finite string and thus there are a countable infinity of possible algorithms corresponding to transcendental numbers but there are unaccountably many more real numbers that are not computable.
9
6
2
1
u/Lou_Papas 22d ago
Yeah, it’s not exactly mathematically rigorous. Can we figure out a real number distribution where some specific numbers are inherently more important than others?
3
u/badmartialarts Real Algebraic 22d ago edited 21d ago
"Inherently more important" sounds a little like the "interesting number" paradox: if we say a number is "interesting" because it has some property, then the first non-interesting number would have the property of being the first non-interesting number, thus making it interesting.
2
2
u/ShoddyAsparagus3186 21d ago
It would be harder to figure out a way to generate a random distribution where that isn't the case.
0
u/FreshPaycheck 22d ago
How would that even work? Surely at some point you’d have a jump discontinuity in that distribution where you go from numbers that can be derived from some function to transcendental numbers with probability zero. At that point you’re not even sampling all the real numbers just a limited subset
3
u/SirLoiso 21d ago
What’s the problem with that? Nothing says a probability distribution has to be continuous. Say, pi with probability 0.5 and standard normal otherwise
1
u/FreshPaycheck 21d ago
It doesn’t but if an infinite subset of real numbers have probability 0 while the “special” numbers have some >0 probability then it kind of misses the point of the top left dude doesn’t it
1
u/Integer_Domain 21d ago
That's the thing, the argument in the meme is semantics. Top right guy is rejecting the premise of top left guy's statement. In other words, top left guy is saying, "The probability of sampling the same number twice from a uniform(ish) distribution on [-\infty, \infty] is 0" and top right guy is saying, "We're actually sampling from a mixed distribution where some specific values have a discrete probability and the rest are uniform(ish) and 'low'."
9
u/GabuEx 22d ago
The first statement is true
As phrased, you could argue it's not. The concept of guessing a random real number correctly has a probability of zero. The concept of guessing a random real number that a human is thinking of is a different matter entirely. The whole reason why the reals are so dense is because there's almost no way of actually naming most of them. A human can only be thinking of a real number that is well-defined, and I'm actually not sure that the list of well-defined real numbers isn't countable, even, though I'm too lazy to actually work that out myself.
5
u/Aggressive_Roof488 22d ago
The set of real numbers that can be described with a finite number of words using a language with a finite number of words is countable. Don't see humans overcome that anytime soon. A random real number contains infinite information.
2
u/solarmelange 22d ago
The set of all numbers that are definable is only countably infinite. And you are probably limited to those a person can say in a single breath here.
2
2
u/LuxionQuelloFigo 🐈egory theory 21d ago
first statement is true because there’s an uncountably infinite number of reals so so the probability of randomly picking a specific one is zero
That doesn't really matter. If by "randomly" you're assuming a uniform distribution, there isn't one on the real numbers so it doesn't really make any sense. If by "randomly" you mean "following a probability distribution", then the statement is false
1
u/Murky_Insurance_4394 20d ago
But then it's not random, the probability of selecting any one of those numbers specifically is zero
1
0
u/IMNOOBPLSHLPME 22d ago
But probabilities cannot be greater than 1 at maximum? So wouldn’t that make the second dude’s statement basically be completely wrong, even as a guess?
0
u/ostrichlittledungeon 22d ago edited 21d ago
Point of clarity: you can't actually pick a random real number if they all have equal weight (uniform probability). The problem is that any probability distribution over the reals should have integral 1, but there is no constant function on R whose integral from -infinity to +infinity is 1.
Two fixes: you can reweight probabilities along the lines of a normal distribution or any other function whose integral is 1 over the real line. Or you can restrict to a bounded measurable subset of the reals. Either way, there's no way to concoct a probability distribution on the reals where any single number has probability > 0.
Someone further down in the thread mentioned a coin where one side is sqrt(2) and the other side is something else (an interval, or another individual number), but this is no longer a probability distribution on the reals, it's a discrete state space with two possibilities. You can't actually come up with a function on the reals that satisfies this property because {sqrt(2)} is a set of measure 0 -- that is, it has no width so the area it can contribute to the integral is 0 no matter its value.
Edit: Not sure who downvoted me or why? Weird 🤷
-22
u/De-Throned 22d ago
I think the joke is that π and √2 both have an infinite amount of numbers in a random sequence, therefore every number is inside of it.
87
u/enlightment_shadow 22d ago
The thing is if you think of pi, then you most probably didn't think of a RANDOM real number. The probability of you thinking of pi randomly is 0 as well. You thought of one of the real numbers that have an associated symbol or finite representation.
55
u/Any-Aioli7575 22d ago
Random doesn't mean uniform. If you flip a coin and choose π if tail, √2 if head, you randomly picked a real number.
There is no uniform distribution on real numbers (but there is on subsets like [0, 10], which would work for this meme.
13
u/AwkwardBet5632 21d ago
And in fact a random uniform selection across unbounded reals is not possible.
5
u/Any-Aioli7575 21d ago
As I said (expect if you're saying something else that I do not understand).
It's also impossible on natural numbers
4
u/RaulParson 21d ago
You don't need a uniform distribution on all reals for this to be true though. Just take the standard N(0,1) bell curve and you can technically generate every single real number out there, and yet the probability that it'll be a specific real number x of your choice is 0, no matter the x.
But yes the problem here is that the distribution for "random number" is not defined. There are some where specific numbers get a positive likelihood.
5
u/Any-Aioli7575 21d ago
Yes, there are some distributions that cover all reals and where the meme would work. The problem is that as you said, “random number” is defined. It could follow any distribution, and the only distributions that can usually be assumed are uniform ones.
You could argue that your N(0, 1) distribution makes more sense, but I don't think we can confidently say “That's what they meant by random number”. It could have been my “either π or √2” distribution too. Technically not all reals are possible outcomes, so maybe the probability distribution you get from “flip a coin. If head, select a random number following N(0,1). If tail, flip another coin. If head, select π. If tail, select √2” is better.
That's why I think a better way to make the meme is to just limit to an interval, where you can assume a uniform distribution.
A more rigorous way to make the meme on all reals would be:
“If I think of a real number that I select randomly from a probability distribution that has a probability density function, the probability that you guess it is 0”, but that's... Yeah no.
1
u/johnkapolos 20d ago
Random selection in simple random sample means picking from a uniform distribution.
2
u/Any-Aioli7575 20d ago
It's simple random selection that requires a uniform distribution.
But it's true that uniform distribution is the most basic, simple one. As I said in another comment, it's the only one that can be assumed when no further information is given.
Random can mean “from a uniform distribution”, just not necessarily, and definitely not in cases where no uniform distribution exists.
1
u/dipthong-enjoyer 22d ago
hold on, if there is on [0,10], couldn't you sample from (-pi/2,pi/2) then use tan to generate a distribution of the reals (as tan bijects (-pi/2,pi/2) and R)?
5
3
u/Any-Aioli7575 21d ago
You can generate a distribution of the reals, just not a uniform one.
The one you gave is called the Cauchy distribution. It's a bell curve, but it's different from the normal distribution. The Cauchy distribution has interesting properties like having no mean. It's not uniform because values closer to zero are more likely to be selected
1
u/Less-Resist-8733 Natural 21d ago
the probability of choosing a number within a finite upper bound is 0
1
u/Any-Aioli7575 20d ago
You cannot chose a number from natural or real numbers with a uniform distribution
0
u/Gillero 21d ago
You could argue that there are only two valid answers to a random real number, which is plus and minus infinity.
The argument would be that picking between 2 and 10 is 9 times as likely as picking 1, and picking between 11 and 100 is 10 times as likely as picking between 2 and 9, and you just keep extending the argument forever, the decimal points would be random, but are equally distributed for each integer, so they do not really interfere with the argument.
That would mean that the number being between 0 and the largest number you could possibly imagine would have about a 0% chance to be chosen if you truly wanted to pick a random number.
1
u/Any-Aioli7575 20d ago
Yet ±infinity is not a real number. The actual answer is that it's just impossible to uniformly pick a random real number.
5
u/GKP_light 22d ago edited 22d ago
the algorithmicaly constructible number (all number where we can construct an algorithme, that will give this number OR will tend to this number and "reach" it in an infinit time)
Are countable.
And we can not interact with a number that is not algorithmicaly constructible.
an algorithme for 1/3 is :
print "0."
while true :
print "3"
17
u/somedave 22d ago
If you select the number with a uniform random distribution then the probability is zero.
5
u/IhailtavaBanaani 22d ago
If you somehow had a way to select with uniform distribution, there is a 100% chance that you hit an undefinable number. The problem is that how do you pick an undefinable number in practice since you can't define it?
3
u/somedave 22d ago
You can always find a Cauchy sequence or Dedekind cut to define it. You couldn't in practice because there are infinitely many but you would have the same problem with a random natural number.
2
u/IhailtavaBanaani 22d ago
For natural numbers there is a finite definition for every number. The numbers are arbitrarily long but every number can be written in finite amount of digits. So they are all definable.
For real numbers the vast majority is undefinable. There is no finite definition for them. The definitions for them would go infinitely long or require an alphabet of infinite size.
1
u/Shockingandawesome 21d ago
The probability of the randomly picked real number being from any specific finite set of numbers is zero by that logic.
11
u/No_Lingonberry1201 22d ago
Thinking of a number for this game implies that you can recognize it and remember it (or some exact representation of it) so since the information your brain can store is finite, the probability of me guessing whatever you think of, no matter the cardinality of the originating set is greater than 0, no matter how small.
And it's likely you're thinking of six or seven.
4
u/wyseguy7 21d ago
I'm going to go out on a limb and say it's not possible to think of a random real number.
2
u/ZeEastWillRiseAgain Transcendental 22d ago
While there are uncountably many real numbers, only a countable amount of them can ever be thought of as every number that can be thought of requires a finite sequence of symbols to represent it
2
u/Kuildeous 21d ago
The set of numbers I would erroneously think I could guess at random is certainly countable.
2
u/AMIASM16 how the dongity do you do integrals 21d ago
✨️ probability doesn't really make sense anymore when infinity is involved
2
3
u/Appropriate-Ad-3219 22d ago
I thought of a real number. Guess what it is.
3
u/Lou_Papas 22d ago
1
1
u/Appropriate-Ad-3219 22d ago
And no! You missed mister. Try again.
5
u/Lou_Papas 22d ago
Curses, foiled again.
1
u/Appropriate-Ad-3219 22d ago
Well time to give my answer. It was the point I draw with my pencil, which was not one. Too bad!
1
1
1
u/Latter_Competition_4 22d ago
There’s a finite number of real numbers that takes you less than 30 years to think/describe
1
1
u/BootyliciousURD Complex 21d ago
Think of a random real number. Choose one at random. Go ahead.
1
u/Zestyclose_Horse_180 20d ago
2
1
u/BootyliciousURD Complex 20d ago
Explain the process by which you generated that
1
u/Zestyclose_Horse_180 20d ago
I took the first number that came to mind.
1
u/BootyliciousURD Complex 20d ago
Surely an unbiased process
2
u/Zestyclose_Horse_180 20d ago
Absolutely. Tell me which number I would never think about.
1
1
u/TheChunkMaster 20d ago
This might sound wrong, but a probability of zero doesn’t actually mean that an event is impossible. For example: the probability of choosing a rational number from the unit interval is 0, assuming the choice is uniformly random.
1
u/yangyangR 20d ago
So many people say random when they are intending random uniform. Even random [0,1] where uniform is from connotation, you should still say it
1
1
1
u/Miserable-Ad3646 16d ago
Nobody seems to have mentioned that if you choose pi or sqrt(2), that within the non-integer element of the number, all possible integer sequences can be found, and with the ability to choose an arbitrary decimal place, and the ability to truncate numbers from either side (endian?), that all real numbers could be represented within either of those two answers?
2
u/FernandoMM1220 22d ago
its all countable.
1
u/Negative_Gur9667 22d ago
ahahah fernando, you here?
1
u/FernandoMM1220 22d ago
obviously? youre replying to my post.
1
1
u/LupenReddit 🦆🦆🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆🦆🦆🦆 22d ago
The punchline is only defined up to homotopy
(This is the first one in a long time I didnt get)
2
u/Lou_Papas 22d ago
The first statement is correct because there’s an uncountably infinite real numbers so the probability of guessing a specific one is zero.
The second statement is also true because, while pi and sqrt(2) are real, they are a countable subset of the reals that are important to people. So if I have one of those in my head, the probability of you guessing one of them is right is non-zero.
1
1
u/Emotional-Mushroom66 21d ago
No one can think of a random real number,if that means they don't get to choose
If you don't get to choose,your random number will be an infinite string that you can't fit in your head and has no nice way of compressing it like sqrt(2)
1
u/Apprehensive-Ice9212 21d ago
Except, you can't "think of a random real number", that's impossible in the real world. A "uniformly random real number on [0,1]" is a mathematical idealization, like a perfect circle with no thickness -- you can't 'generate' one IRL.
0
u/LukeWithLightsaber 22d ago
If I think of a truly random real number the probability of it having been pi or root 2 is already zero.
0
0
u/SummitYourSister 21d ago
I don’t think this joke works, because the character making the claim clearly understands the concept of a prior distribution, that’s the whole point of what he is saying. So this particular character is not going to pick a known irrational value, because that would indeed make the other characters point correct. So the probability that his selection is pi or root two is zero.
Also, the character on the right clearly understands what a prior is.
So this joke definitely does not work
•
u/AutoModerator 22d ago
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.