Geometrical probability

Pick two points A and B independently at random on the circumference of a circle $C$ with centre $O$ and unit radius. Let $\Pi$ be the length of the perpendicular from $O$ to the line $A B$, and let $\Theta$ be the angle $A B$ makes with the horizontal. Show that $(\Pi , \Theta )$ has joint density

Let $S_{1}$ and $S_{2}$ be disjoint convex shapes with boundaries of length $b\left(S_{1}\right), b\left(S_{2}\right)$, as illustrated in the figure beneath. Let $b(H)$ be the length of the boundary of the convex hull of $S_{1}$ and $S_{2}$, incorporating their exterior tangents, and $b(X)$ the length of the crossing curve using the interior tangents to loop round $S_{1}$ and $S_{2}$. Show that the probability that a random line crossing $S_{1}$ also crosses $S_{2}$ is $\left\{ b(X)-b(H)\right\} / b\left(S_{1}\right)$. (See Example (4.13.2) for an explanation of the term 'random line'.) How is this altered if $S_{1}$ and $S_{2}$ are not disjoint?

The circles are the shapes $S_{1}$ and $S_{2}$. The shaded regions are denoted $A$ and $B$, and $b(X)$ is the sum of the perimeter lengths of $A$ and $B$.

Let $S_{1}$ and $S_{2}$ be convex figures such that $S_{2} \subseteq S_{1}$. Show that the probability that two independent random lines $\lambda_{1}$ and $\lambda_{2}$, crossing $S_{1}$, meet within $S_{2}$ is $2 \pi \left|S_{2}\right| / b\left(S_{1}\right)^{2}$, where $\left|S_{2}\right|$ is the area of $S_{2}$ and $b\left(S_{1}\right)$ is the length of the boundary of $S_{1}$. (See Example (4.13.2) for an explanation of the term 'random line'.)

Let $Z$ be the distance between two points picked independently at random in a disk of radius $a$. Show that $\mathbb {E}\left[Z\right] = 128 a /(45 \pi )$, and $\mathbb {E}\left[Z^{2}\right] = a^{2}$.

Pick two points $A$ and $B$ independently at random in a ball with centre $O$. Show that the probability that the angle $\widehat{\mathrm{AOB}}$ is obtuse is $\frac{5}{8}$. Compare this with the corresponding result for two points picked at random in a circle.

A triangle is formed by $\mathrm{A}, \mathrm{B}$, and a point P picked at random in a set $S$ with centre of gravity G. Show that $\mathbb {E}\left[\left|\mathrm{ABP}\right|\right] = \left|\mathrm{ABG}\right|$.

A point $D$ is fixed on the side $B C$ of the triangle $A B C$. Two points $P$ and $Q$ are picked independently at random in $A B D$ and $A D C$ respectively. Show that $\mathbb {E}\left[\left|A P Q\right|\right] = \left|\mathrm{AG}_{1} \mathrm{G}_{2}\right| = \frac{2}{9}\left|A B C\right|$, where $G_{1}$ and $\mathrm{G}_{2}$ are the centres of gravity of ABD and ADC.

From the set of all triangles that are similar to the triangle $A B C$, similarly oriented, and inside ABC, one is selected uniformly at random. Show that its mean area is $\frac{1}{10}\left|\mathrm{ABC}\right|$.

Two points $X$ and $Y$ are picked independently at random in the interval $(0, a)$. By varying $a$, show that $F(z, a) = \mathbb {P}\left(|X-Y| \leq z\right)$ satisfies

and hence find $F(z, a)$. Let $r \geq 1$, and show that $m_{r}(a) = \mathbb {E}\left[|X-Y|^{r}\right]$ satisfies

Hence find $m_{r}(a)$.

Lines are laid down independently at random on the plane, dividing it into polygons. Show that the average number of sides of this set of polygons is 4. [Hint: Consider $n$ random great circles of a sphere of radius $R$; then let $R$ and $n$ increase.]

A point $P$ is picked at random in the triangle $A B C$. The lines $A P, B P, C P$, produced, meet $B C$, $A C, A B$ respectively at $L, M, N$. Show that $\mathbb {E}\left[\left|L M N\right|\right] = \left(10-\pi^{2}\right)\left|A B C\right|$.

If four points are picked independently at random inside the triangle $A B C$, show that the probability that no one of them lies inside the triangle formed by the other three is $\frac{2}{3}$.

If three points $P, Q, R$ are picked independently at random in a disk of radius $a$, show that $\mathbb {E}\left[\left|\mathrm{PQR}\right|\right] = 35 a^{2} /(48 \pi )$. [You may find it useful that $\int_{0}^{\pi } \int_{0}^{\pi } \sin^{3} x \sin^{3} y \sin \left|x-y\right| d x d y = 35 \pi / 128$.]

Two points $A$ and $B$ are picked independently at random inside a disk $C$. Show that the probability that the circle having centre A and radius $\left|\mathrm{AB}\right|$ lies inside $C$ is $\frac{1}{6}$.

Two points A and B are picked independently at random inside a ball $S$. Show that the probability that the sphere having centre A and radius $\left|\mathrm{AB}\right|$ lies inside $S$ is $\frac{1}{20}$.

Pick two points independently and uniformly at random on the surface of the sphere of $\mathbb {R}^{3}$ with radius 1. Show that the density of the Euclidean distance $D$ between them is $f(d) = \frac{1}{2} d$ for $d \in (0,2)$.

Two points are chosen independently and uniformly at random on the perimeter (including the diameter) of a semicircle with unit radius. What is the probability of the event $D$ that exactly one of them lies on the diameter, and of the event $N$ that neither lies on the diameter?

Let $A$ be the area of the triangle formed by the two points and the midpoint of the diameter. Show that $\mathbb {E}\left[A \mid D\right] = 1 /(2 \pi )$, and $\mathbb {E}\left[A \mid N\right] = 1 / \pi$. Hence or otherwise show that $\mathbb {E}\left[A\right] = 1 /(2+\pi )$.

Stevens's solution of Jeffreys's bicycle wheel problem. After passing over a stretch of road strewn with tacks, a cyclist looks repeatedly at the front wheel to check whether the tyre has picked up a tack. One inspection of the wheel covers a fraction $x$ of the tyre. After $n$ independent such inspections, each uniformly positioned on the wheel, show that the probability of having inspected the entire tyre is $1-\binom {n}{1}(1-x)^{n-1}+\binom {n}{2}(1-2 x)^{n-1}-\cdots$, where the series terminates at the term in $1-k x$ where $k = \lfloor 1 / x\rfloor$ is the integer part of $1 / x$.