# Questions tagged [geometric-probability]

Probabilities of random geometric objects having certain properties (enclosing the origin, having an acute angle,...); expected counts, areas, ... of random geometric objects. For questions about the geometric distribution, use (probability-distributions) instead.

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### Measure transport by a random matrix

I want to understand what happens to a measure on $\mathbb{R}^n$ when it is transported by a random matrix. The idea is that I want to pick random vectors, apply a random matrix, and see how they are ...
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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 ...
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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 ...
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### Conjecture: Two different random triangles (both based on random points on a circle) have the same distribution of side length ratios.

On a circle, choose three uniformly random points $A,B,C$. Triangle $T_1$ has vertices $A,B,C$. The side lengths of $T_1$ are, in random order, $a,b,c$. Triangle $T_2$ is formed by drawing tangents to ...
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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 ...
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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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### Probability that the centroid of a triangle is inside its incircle

Question: The vertices of triangles are uniformly distributed on the circumference of a circle. What is the probability that the centroid is inside the incricle. Simulations with $10^{10}$ trails ...
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### The probability of a circle in a circumscriptible polygon

I have difficulty understanding the solution below and have already summarized my difficulties as follows, why "the area of the polygon $abcelef$ .... represents the number of ways the three ...
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### Probability of each type of inscribed octahedron

Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. ...
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### Probability that the coefficients of a quadratic equation with real roots form a triangle

Question: What is the probability that the coefficients of a quadratic equation form the sides of triangle given that it has real roots? Assume that the coefficients are uniformly distributed and ...
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### 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 ...
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### Probability that a triangle inscribed in a square comprises at least $\frac{1}{4}$ of the area of the square

Question: Suppose that points $P_1$, $P_2$, and $P_3$ are chosen uniformly at random on the sides of a square $T$. Compute the probability that $$\frac{[\triangle P_1 P_2 P_3]}{[T]}>\frac{1}{4}$$ ...
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### Probability Theory: Generating Functions of Random Variables

Let $X, Y$ be independent random variables with the geometric distribution with parameter $p > 0$. (a) Compute the mean of $Z = XY$. I got that $E(Z) = 1/p^2$ (b) Compute the probability ...
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### Expected radius of throwing a dart at a dartboard

I am doing a problem that states: If you are throwing a dart at a circular board with radius $R$, what is the expected distance from the centre? If $x$ is the expected radius, then it would be the ...
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### Distribution of a combination of four uniformly distributed variables: $X_1+X_2 +\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$

My problem involves four random variables $X_1, Y_1, X_2, Y_2 \sim U(0,1)$ in the expression $Z = X_1 + X_2 + \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}$. From what I understand so far, I need to find the ...
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### $n\times n$ grid filled with $n$ colors. What is the average group size as $n\to\infty$

Take a grid with dimensions $n\times n$ squares and randomly fill each square with $1$ of $n$ colors. What is the expected average group size of colors touching each other as $n$ approaches $\infty$? ...
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### Three random points on $x^2+y^2=1$ are the vertices of a triangle. Is the probability that $(0,0)$ is inside the triangle's incircle exactly $0.13$?

Three uniformly random points on the circle $x^2+y^2=1$ are the vertices of a triangle. What is the probability that $(0,0)$ is inside the triangle's incircle? (This a variation of the question &...
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### A square contains many random points. From each point, a disc grows until it hits another disc. What proportion of the square is covered by the discs?

A square lamina contains $n$ independent uniformly random points. At a given time, each point becomes the centre of a disc whose radius grows from $0$, at say $1$ cm per second, and stops growing when ...
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### How to define pdf of the distance to the point of the spherical cap?

Suppose we have a sphere centered at $(0, 0, 0)$, with the radius of $R_b$. We cut the sphere with the tangent plane centered at $(0, 0, R_a)$, where a dude is fixed on. (Here $0 < R_a < R_b$, ...
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### A disc contains $n$ random points. Each point is connected to its nearest neighbor. What does the average cluster size approach as $n\to\infty$?

A disc contains $n$ independent uniformly random points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random ...
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### Buffon's needle in one dimension

I want to solve Buffon's Needle problem but first I was trying to tackle a simpler case. So: consider an infinite line with points each $t$ units. Let's say that we have a "needle" of length ...
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### Probability that two random lines intersect inside a square

Consider the square with vertices $(0,0),(1,0),(1,1),(0,1)$. Choose two independent uniformly random points $P$ and $Q$ inside the square. Draw a line $l_P$ connecting $(0,0)$ and $P$. Draw another ...
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