All Questions
Tagged with geometric-probability probability
444
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Probability of 3 darts landing in the same half of the board [duplicate]
Problem: Find the probability of 3 randomly thrown darts landing in the same half of the board.
More generally, if $n$ points picked uniformly randomly on a disk, find the probability of them lying in ...
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1
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Expected number and variance of number when sum exceeds 1 (Irwin-Hall distribution)
I am given a random variable (uniformly distributed) between 0 and 1. To this, I add a second such random variable. I keep on adding these variables until sum exceeds 1, and then stop. Let us call ...
- 342
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3
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A mysterious limit: probability that a triangle captures the centre of a circle.
On a circle, choose $6n$ $(n\in\mathbb{Z^+})$ uniformly random points and label them $a_0,a_1,a_2,\dots,a_{6n-1}$ going anticlockwise, with $a_0$ chosen randomly.
Draw three chords:
Chord $a_0 a_{3n}$...
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Is there anyway to guarantee probability mass coverage?
If I have a probability density function on $\mathbb{R}^n$. I can sample $m$ points from it. Is there anyway to get an estimate of how much probability mass is covered by balls of radius $\delta$ ...
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Probability between a rectangular 2D die and a squared 2D die
I'm trying to find a solution to the next problem:
If I have a rectangular $2D$-die with uniform density such that each side has a certain probability $P1,P2,P3,P4$ respectively.
I want to find a ...
- 1
9
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1
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A probability involving side lengths of a random triangle on a disk: Is it really $\frac37$?
Choose three uniformly random points on a disk, and let them be the vertices of a triangle. Call the side lengths, in random order, $a,b,c$.
What is $P(a^2<bc)$ ?
A simulation with $10^7$ such ...
- 25.7k
16
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1
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A probability involving areas in a random pentagram inscribed in a circle: Is it really just $\frac12$?
The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. The petals are randomly chosen, but they must be ...
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The vertices of a pentagram are five random points on a circle. Conjecture: The probability that the pentagram contains the circle's centre is $3/8$.
The vertices of a pentagram are five uniformly random points on a circle.
Is the following conjecture true: The probability that the pentagram contains the circle's centre is $\frac38$.
(The ...
- 25.7k
15
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2
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The vertices of a hexagon are random points on a unit circle; $a,b,c$ are the lengths of three random sides. Conjecture: $P(ab<c)=\frac35$.
The vertices of a hexagon are uniformly random points on a unit circle; $a,b,c$ are the lengths of three distinct random sides.
A simulation with $10^7$ such random hexagons yielded a proportion of $0....
- 25.7k
9
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Probability that Mercury is the nearest planet to Earth.
Motivation: We tend to think of Venus as the nearest planet to Earth because at its nearest approach to Earth, Venus is the closest at 39 million Km away. This is followed by Mars at 56 million Km and ...
- 20.8k
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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
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Probability of two geometric conditions happening together
I have a problem combining geometry with probability, and I feel like I do not understand the basics to approach this problem.
Let there be two points $\mathbf{p}_1$ and $\mathbf{p}_2$ in $\mathbb{R}^...
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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
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4
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Probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation.
My question is: What are some examples of probability questions that have answer $\frac{1}{2}$ but resist intuitive explanation?
Context
Some probability questions have answer $\frac{1}{2}$, and - as ...
- 25.7k
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4
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Draw tangents at 3 random points on a circle to form a triangle. Show that the probability that a random side is shorter than the diameter is $1/2$.
Choose three uniformly random points on a circle, and draw tangents to the circle at those points to form a triangle. (The triangle may or may not contain the circle.) For example:
What is the ...
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