Skip to main content

New answers tagged

1 vote

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

Here is an elementary proof using basic trigonometry for the more general formula $p(ab<rc)=\frac{2 \arctan(r)}{\pi}.$ Fixing one point on a circle of radius $r$ and placing the other two at angles ...
Ivan Meir's user avatar
  • 4,792
6 votes
Accepted

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

One can use basic probability theory to prove that $$P(ab<kc)=\frac{2}{\pi}\arctan k,\qquad k>0.$$ Without loss of generality, the vertices opposite the sides $a,b,c$ are $$A=e^{2i\beta},\qquad ...
GH from MO's user avatar
  • 99.4k
5 votes

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

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 $...
Dan's user avatar
  • 2,837
6 votes

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

The following argument is less or more the same as that of Iosif Pinelis, but with less computations and the symmetry is rather explicit. It may be explained without complex numbers, but the ...
Fedor Petrov's user avatar
2 votes

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

Denoting the angles corresponding to edges of side lengths $a,b,c$ with $A,B,C$ respectively, by the sine law: $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2.$$ Thus $c>ab$ amounts to $\sin(...
KhashF's user avatar
  • 2,857
0 votes

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

We use the formula (from en.wikipedia.org/wiki/Circumcircle#Other_properties ): diameter = a * b * c / ( 2 * area ) In our case diameter = 2 and we get: 4 * area = a * b * c If h is the height of ...
Patrik's user avatar
  • 614
5 votes

The vertices of a triangle are three random points on a unit circle. The side lengths are, in random order, $a,b,c$. Show that $P(ab>c)=\frac12$

Without loss of generality, the three points on the unit circle are $1,e^{is},e^{it}$, where $s$ and $t$ are uniformly and independently distributed on the interval $[0,2\pi]$, so that $a=|e^{is}-1|$, ...
Iosif Pinelis's user avatar
0 votes

Random partition of an interval – Dirichlet distributed?

Suppose $1\le k_1\le k_2\le \cdots \le k_\ell \le N$ are integers. Then the probability distribution of the vector of lengths of $$ [0, X_{(k_1)}], [X_{(k_1}, X_{(k_2)}], \ldots, [X_{(k_\ell)}, 1] $$ ...
Michael Hardy's user avatar
1 vote

Random partition of an interval – Dirichlet distributed?

This follows from the properties of the Poisson process. Consider a Poisson process (say, of rate 1), and let its first $N+1$ points be $Y_1<Y_2<\dots<Y_{N+1}=:A$. Then $\xi_0:=Y_1$, $\xi_1:=...
van der Wolf's user avatar

Top 50 recent answers are included