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A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$. Let's tweak this question by making each random numbe …

This question resisted attacks at MSE, so I am posting it here. 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 …

This question was posted on MSE. It received some interesting responses, but no definite answer. Let's build a variation of Pascal's triangle. We write $1$'s going down the sides, as usual. Then for …

On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random point on the bottom disk …

A disc contains $n$ independent uniformly distributed points. Each point is connected by a line segment to its nearest neighbor, forming clusters of connected points. For example, here are $20$ random …

This question was posted at MSE but was not answered. 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 pet …

The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$. There is a convoluted proof that $P(ab>c)=\frac12$. But since the probabil …

I think the comment by @James Martin answers my question. The number of clusters equals the number of pairs of points that are mutually nearest neighbors. The probability that a point is in a mutually …

Consider the graph of $\sin^p x+\sin^p y=\sin^p(x+y)$, where $x$ and $y$ are acute, and $p>1$. Here are examples with, from left to right, $p=1.05,\space 1.25,\space 2,\space 4,\space 100$. Find the …

A proof from pure geometry On a unit circle with centre $O$, draw parallel chords $PQ$ and $P'Q'$ such that $PQ'\perp P'Q$. Chord $MN$ is parallel to $PQ$ and passes through $R$, the intersection of $ …

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is …

The question has been answered at the original MSE question. (A simulation with a larger sample size shows that a $99.9$% confidence interval for the probability is $0.49994\pm0.00002$.)