Articles of geometric probability

Randomly dropping needles in a circle?

If we were to randomly drop $n$ needles of random length in a circle, what would be the odds of finding $k$ intersections? This can be asked as: Randomly place $n$ line segments in a circle. Their length and position is determined by $2$ random points uniformly and independently set in that circle. What are […]

Three points on sides of equilateral triangle

Let’s choose three points on the sides of an equilateral triangle(one point on each side) and construct a triangle with these three points. what is the probability that area of this triangle be at least one half of the area of equilateral triangle?

What is the average length of 2 points on a circle, with generalizations

I have earlier seen the question about finding the average length of two points and $n$ points inside the unit disk. But what about the more simple question, what happens if the points lie exactly on the circle? I did some basic algebra, assume that the radius of the circle is $r$. without loss of […]

Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn’t been adressed, so I’m posting it as a separate question: What is the probability that randomly cutting an equilateral triangle will allow one part […]

Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn’t, you can shift it to the centre while taking from one part and giving to the other.) However, if you […]

What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?

The vertices are chosen completely randomly and all lie on the circumference. Is there a formula for the chance that an $n$-gon covers over $50$% of the area of the circle, with any input $n$? I tried to find something, however I did not know what to look for when I realized the first three […]

Expected area of the intersection of two circles

If we pick randomly two points inside a circle centred at $O$ with radius $R$, and draw two circles centred at the two points with radius equal to the distance between them, what is the expected area of the intersection of the two cirlces that contain the origin $O$.

How was “Number of ways of arranging n chords on a circle with k simple intersections” solved?

The problem whose solution is based on the solution to the problem in the title came up as I was trying to find a simpler variant of my needle problem. I we were to uniformly, randomly and independently set $2n$ points on a circle, and then randomly connect them in a way such that each […]

Probability of the Center of a Square Being Contained in A Triangle With Vertices on its Boundary

Background : I happen to love solving tough problems. Problem is, I simply cannot answer some! It happened again today, as I attempted to solve the questions in this site: This site seems to have really, really difficult questions, and in fact, I’m struggling to answer the second question! That struggle is not without […]

Probability that 3 points in a plane form a triangle

This question was asked in a test and I got it right. The answer key gives $\frac12$. Problem: If 3 distinct points are chosen on a plane, find the probability that they form a triangle. Attempt 1: The 3rd point will either be collinear or non-collinear with the other 2 points. Hence the probability is […]