# Category Archives: Probability

## Sphere

Pick randomly three points on the unit sphere. The geodesic triangle with these points as vertices divides the sphere into two parts. What is the probability that the north pole is contained in the smaller of these parts? For instance … Continue reading

## Prisoner’s problem 2

Please comment your solutions, questions and remarks.. Consider a prison with several millions prisoners. The head of the prison arranges the following challenge to the prisoners. One prisoner at a time will be called to a room with a lamp. … Continue reading

## Andre’s Reflection Method

I promised to post something about random walks. I learned the content of this post while listening to Vilppu Tarvainen’s bachelor’s degree talk and I consider it is quite cute. Let us consider a one dimensional random walk: formally it … Continue reading

Posted in Combinatorics, Mathematics, Probability, Recreation | 1 Comment

## Continuum Hypothesis III

Suppose you pick randomly a real number. What is the probability that it equals to 1? The probability is zero. Suppose $$X\subset [0,1]$$ is a countable subset of the unit interval. What is the probability that a randomly picked real … Continue reading

Rami Luisto asked the following question: Pick randomly n unit vectors from the plane, $$x_1,\dots,x_n\in \mathbb{R}^2$$. What is the probability that the sum of these vectors has magnitude less than or equal to one, $$|x_1+\cdots+x_n|\leqslant 1$$? (magnitude = norm = … Continue reading