
Buffon's needle or how ants estimate nest size - billconan
https://epiphany.pub/post?refId=9977c7ea88099bc6d6e607e4477f3920eb4440ca781953b5bdc2c6f92e19f739
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samsamoa
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](https://en.wikipedia.org/wiki/Crofton_formula)

[1]
[https://arxiv.org/pdf/1505.05515.pdf](https://arxiv.org/pdf/1505.05515.pdf)

~~~
srean
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)

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malisper
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)](https://en.wikipedia.org/wiki/Bertrand_paradox_\(probability\))

~~~
CDSlice
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.

~~~
malisper
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.

~~~
OscarCunningham
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.

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runeblaze
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.

~~~
quickthrower2
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.

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will_pseudonym
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.soa.org/education/exam-req/edu-exam-p-detail.aspx)

[https://www.casact.org/admissions/syllabus/index.cfm?fa=1syl...](https://www.casact.org/admissions/syllabus/index.cfm?fa=1syllabi)

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lorenzhs
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.

~~~
targonca
>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.

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safgasCVS
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

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lonelappde
Kakeya is another nice needle problem:

[https://youtu.be/IM-n9c-ARHU](https://youtu.be/IM-n9c-ARHU)

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Rainymood
Unreadable on Chrome.

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billconan
Which platform? Mobile? Which version?

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Rainymood
Weird. It is fixed now. It got stuck loading something I think and didn't
expand beyond the first paragraph. Now it does. My bad.

