All Questions
Tagged with geometric-probability definite-integrals
9
questions
5
votes
1
answer
77
views
Probability of a random cyclic quadrilateral enclosing a fixed point in its circle
I finally found a single integral solving the natural generalisation of the problem discussed here:
For $n\ge1$ pick $n+2$ points uniformly at random on the unit circle. What is the probability $P_n(...
- 104k
22
votes
4
answers
1k
views
Find the area of the region enclosed by $\frac{\sin x}{\sin y}=\frac{\sin x+\sin y}{\sin(x+y)}$ and the $x$-axis.
Here is the graph of $\dfrac{\sin x}{\sin y}=\dfrac{\sin x+\sin y}{\sin(x+y)}$.
Find the area of the region enclosed by the curve and the $x$-axis, from $x=0$ to $x=\pi$.
Where the question came ...
- 25.7k
16
votes
2
answers
829
views
The vertices of a triangle are three random points on a unit circle. The side lengths are $a,b,c$. Show that $P(ab>c)=\frac12$.
The vertices of a triangle are three uniformly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
Show that $P(ab>c)=\frac12$.
The result is strongly suggested by ...
- 25.7k
6
votes
3
answers
244
views
What is the expected length of the hypotenuse formed by bending a unit length randomly at a right angle?
This is easy enough to simulate and find the answer is somewhere around .812. However I am not finding it so easy to solve the integral involved which I believe is...
$$\int_0^1 \sqrt{x^2+(1-x)^2} dx$...
- 1,024
4
votes
2
answers
721
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Probability of euclidean distance between two random points inside a unit circle/sphere greater than 1
Problem: Say there are two points inside the circle; A and B, and they are both randomly drawn according to a uniform ...
- 143
10
votes
2
answers
670
views
A surprising dilogarithm integral identity arising from a generalised point enclosure problem
This question asked:
What is the probability that three points selected uniformly randomly on the unit circle contain a fixed point at distance $x$ from the circle's centre?
I answered that ...
- 104k
7
votes
1
answer
347
views
Area bounded by $\cos x+\cos y=1$
What is the area of the region $\cos x+\cos y > 1$, where $|x|,|y|<\pi$?
In other words, is there a "closed" form -- using functions that are well-known and nice to work with -- for this ...
- 27k
0
votes
1
answer
589
views
Geometric probabilities with rectangle
One side of rectangle is 1.2 other is 3.9. We randomly pick points on adjacent sides and then draw a stretch through them. What is the probability that the area of the received triangle is less than 1....
- 301
1
vote
0
answers
49
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Question regarding double integrals
Regarding the Buffon's needle case for long needles of length $ l>t, $ (the distance between the parallel lines on the floor), we need to solve the integral $$ \int_{\theta=0}^{\frac{\pi}{2}} \int_{...
