Buffon's needle or how ants estimate nest size(epiphany.pub)
epiphany.pub
Buffon's needle or how ants estimate nest size
https://epiphany.pub/post?refId=9977c7ea88099bc6d6e607e4477f3920eb4440ca781953b5bdc2c6f92e19f739
27 comments
I thought this was total magic in high school until my college's intro to probability class discussed this before the midterm. I guess maths progression in college can be fast.
I think high school is constrained by having to normalize the curriculum over the entire population, but college (I take that to mean university) has more leeway to teach you what they like, and make it harder. Plus you are only studying 1-2 subjects.
[deleted]
There’s a nice chapter on Buffon’s needle problem in “Proofs from THE BOOK”, an Erdős-inspired attempt to collect the most beautiful proofs. You can find a PDF of the entire book easily enough on Google, it’s a good read although I would dispute the beauty of some of the proofs in it.
>although I would dispute the beauty of some of the proofs in it
Agree. In my opinion, the proofs in that book are not so much about beauty, rather how one can prove seemingly complex stuff with surprisingly elementary tools.
Agree. In my opinion, the proofs in that book are not so much about beauty, rather how one can prove seemingly complex stuff with surprisingly elementary tools.
Unreadable on Chrome.
This was a practice problem I remember from when I was studying for Exam 1/P, one of the qualifying exams for actuaries.
https://www.soa.org/education/exam-req/edu-exam-p-detail.asp...
https://www.casact.org/admissions/syllabus/index.cfm?fa=1syl...
https://www.soa.org/education/exam-req/edu-exam-p-detail.asp...
https://www.casact.org/admissions/syllabus/index.cfm?fa=1syl...
An interesting related problem is Bertrand's paradox[0]. You have an equilateral triangle inscribed in a circle. You pick a random chord from the circle (a line that goes from one edge of the circle to another). What's the probability the chord is longer than an edge of the triangle?
As it turns out, there isn't a well-defined solution. That's the paradox. The issue comes from "picking a random chord". There are different ways that you can pick a random chord which lead to different solutions. One approach to picking a random chord is to pick two random points on the circle and draw the chord between them. This gives you a probability of 1/3. Another approach is to pick a random direction and radius, then draw a chord at the end of the radius perpendicular to the radius. This approach gives you a probability of 1/2.
One interesting aspect is the second approach is the only approach where if you were to draw a circle within the larger circle, the distribution of the chords in the inner circle will match the distribution of chords in the outer circle.
[0] https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
As it turns out, there isn't a well-defined solution. That's the paradox. The issue comes from "picking a random chord". There are different ways that you can pick a random chord which lead to different solutions. One approach to picking a random chord is to pick two random points on the circle and draw the chord between them. This gives you a probability of 1/3. Another approach is to pick a random direction and radius, then draw a chord at the end of the radius perpendicular to the radius. This approach gives you a probability of 1/2.
One interesting aspect is the second approach is the only approach where if you were to draw a circle within the larger circle, the distribution of the chords in the inner circle will match the distribution of chords in the outer circle.
[0] https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
I am not sure what makes this a paradox? Normally I think of a paradox as leading to a state where you have two or more contradictory statements being true at like Russell's Paradox in set theory. This just seems like a problem where you don't have enough information to solve it but once you get the info you need (how to pick random chords) you get a logically consistent solution.
A paradox is any statement that appears to contradict itself. That includes both actual contradictions such as Russel's paradox and statements that appear to be invalid, but have non-obvious valid conclusions like the Monty Hall problem.
This problems falls into the second category. We are given what appears to be a well-defined problem, but when we solve it in different ways, we get different answers. The resolution to the paradox is the problem isn't well defined because your technique for generating randomness will produce different answers.
This problems falls into the second category. We are given what appears to be a well-defined problem, but when we solve it in different ways, we get different answers. The resolution to the paradox is the problem isn't well defined because your technique for generating randomness will produce different answers.
Because of the Principle of Explosion, a contradiction implies any statement. So for people who don't think that everything is meaningless there can't be any true contradictions. This means that in fact every paradox falls into the second class. Paradoxes merely differ in how difficult it is to see the logical flaw that leads to the apparent contradiction.
It's a paradox because esteemed mathematicians, including authors of classic books, have vociferously argued that their particular arbitrary choice is the sole correct choice.
In probability, one must make many unstated assumptions to get an answer to a problem. Usually one believes those assumptions are "canonical", like uniform independent distributions. Sometimes woldviews contradict each other.
In probability, one must make many unstated assumptions to get an answer to a problem. Usually one believes those assumptions are "canonical", like uniform independent distributions. Sometimes woldviews contradict each other.
[deleted]
Yeah. I'm ok with calling anything that looks weird a "paradox" (even stuff like "The Monty Hall problem"), but in this case describing the problem this way is precisely what's misleading about it: the person describing the "problem" must first pretend that there is a well defined (even though possibly not known to the less educated listener than himself) "random" way to pick 2 points on a circle, which there clearly isn't: there are different algorithms to do that, that produce different distribution, which very obviously look different when plotted, and choosing any specific algorithm/distribution makes the "paradox" disappear. So I really dislike when stuff like that is called a "paradox". It's just a way to trick the listener (or yourself) into thinking there is a problem, when there really isn't any.
> Yeah. I'm ok with calling anything that looks weird a "paradox" (even stuff like "The Monty Hall problem"), but in this case describing the problem this way is precisely what's misleading about it: the person describing the "problem" must first pretend that there is a well defined (even though possibly not known to the less educated listener than himself) "random" way to pick 2 points on a circle, which there clearly isn't: there are different algorithms to do that, that produce different distribution, which very obviously look different when plotted, and choosing any specific algorithm/distribution makes the "paradox" disappear. So I really dislike when stuff like that is called a "paradox". It's just a way to trick the listener (or yourself) into thinking there is a problem, when there really isn't any.
It is also a good way to broach a discussion on the nature of randomness.
It is also a good way to broach a discussion on the nature of randomness.
Exactly, this is just a definition problem
It's never really meaningful to ask "what's the probability of...?" without specifying the probability model with respect to which the question is asked, in this case the assumed sampling distribution of chords.
A beautiful related result is the Crofton Formula [0], which says that you can measure the length of any curve by appropriately counting the number of straight lines that intersect it.
(My co-authors and I discovered a generalization of the formula that also holds in curved space [1].)
[0] https://en.wikipedia.org/wiki/Crofton_formula
[1] https://arxiv.org/pdf/1505.05515.pdf
(My co-authors and I discovered a generalization of the formula that also holds in curved space [1].)
[0] https://en.wikipedia.org/wiki/Crofton_formula
[1] https://arxiv.org/pdf/1505.05515.pdf
Interesting as the paper is, please link to the arxiv abstract rather than the pdf as that's more web friendly. Reader can always click on he pdf link there.
https://arxiv.org/abs/1505.05515
https://arxiv.org/abs/1505.05515
I'm especially intrigued by the "Timelike, Spacelike, and Lightlike" diagram on Page 10 of the linked paper where "lightlike" phenomena share an endpoint. Fascinating.
[deleted]
I first saw this little gem in a stats test in my second year at university. I must say it was a lot more fun deriving it without the time pressure
Kakeya is another nice needle problem:
https://youtu.be/IM-n9c-ARHU
https://youtu.be/IM-n9c-ARHU