Intereting Posts

Proof of the right and left cancellation laws for Groups
Distribution of sums
Continuous and bounded imply uniform continuity?
Verification for the solution following differential equation!
Rigorous Proof: Circle cannot be embedded into the the real line!
Full flag $Fl_{\mathbb C}(3)$
Probability that the convex hull of random points contains sphere's center
When the quadratic formula has square root of zero, how to proceed?
Indefinite summation of polynomials
Question about a proof about singular cardinals
Diffucult Tautology to Prove
Calculus “Word Problem”
Minimal polynomials and cyclic subspaces
An inequality: $1+\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac53$
Lipschitz continuity of atomless measures

For recreational purpose, i haven’t seen a interesting elemetary probability question quite a while.

Is there any surprising elementary probability problem that result in surprising solution like the Monte Hall problem? Please give a few examples.

- Why are rings called rings?
- Is it to the students' advantage to learn the language of infinitesimals?
- Origins of the modern definition of topology
- Why are modules called modules?
- What's the history of the result that $p_{n+1} < p_n^2$, and how difficult is the proof?
- A place to learn about math etymology?

- Expectation value of a product of an Ito integral and a function of a Brownian motion
- Monte-Carlo simulation with sampling from uniform distribution
- Rumors told between $n+1$ people
- When was Regularity/Foundation universally adopted?
- vaccine success CDF
- Binary matrices and probability
- Gaussian distribution on a $2$-sphere
- Expected Value of Max of IID Variables
- Expected number of trials before I get one of each type
- very simple conditional probability question

I think my favorite, for pedagogical purposes, is this deceptively simple one: What is the probability that a random chord chosen in a unit circle is longer than the side of an equilateral triangle inscribed in that circle? It’s an easy question to ask, but the answer — well, the answer is 1/3rd: consider aligning the equilateral triangle such that a vertex is at one end of the chord; then the chord is longer than a side of the triangle iff its other endpoint is in the 1/3rd of the circumference between the other two vertices.

Or maybe the probability is 1/2: imagine choosing a diameter of the circle orthogonal to the chord; then the chord is longer than a side of the triangle iff its point of intersection with the diameter is closer to the center than to the circumference of the circle.

Or maybe it’s 1/4? The chord is longer than an equilateral triangle side iff its midpoint is in the circle of radius 1/2 centered in the unit circle, and that circle has area 1/4 the area of the unit circle.

The catch here is that ‘random chord’ is an ill-defined concept in and of itself; the probability in question is contingent on the distribution of the chords, and each of these probabilities corresponds to a different meaning for the phrase ‘random chord’. For more details, see the Wikipedia entry on Bertrand’s Paradox.

Although an answer has been accepted, I would like to share a problem.

This is an online article that I happened to read in “The Best Writing on Mathematics 2010”:

Knowing When to Stop: How to gamble if you must—the mathematics of optimal stopping

It talks about a few problems including The Marriage Problem, but most notably it also contain this problem that I find very surprising:

“Suppose you must choose between only two slips of paper or two cards. You turn one over, observe a number there and then must judge whether it is larger than the hidden number on the second. The surprising claim, originating with David Blackwell of the University of California, Berkeley, is that you can win at this game more than half the time.”

A solution:

“Here is one stopping rule that guarantees winning more than half the time. First, generate a random number R according to a standard Gaussian (bell-shaped) curve by using a computer or other device. Then turn over one of the slips of paper and observe its number. If R is larger than the observed number, continue and turn over the second card. If R is smaller, quit with the number observed on the first card. How can such a simple-minded strategy guarantee a win more than half the time?”

For the proof I suggest reading the article, as it contains some nice pictures too.

- Finding $\pi$ factorial
- Integer $k$ such that $k!$ has 99 zeros
- Computing the Integral $\int\tanh\cos\beta x\,dx$
- Can Peirce's Law be proven without contradiction?
- Is there other methods to evaluate $\int_1^{\infty}\frac{\{x\}-\frac{1}{2}}{x}\ \mathrm dx$?
- proof of Poisson formula by T. Tao
- what exactly is mathematical rigor?
- Show that a locally compact Hausdorff space is regular.
- Give an algorithm that computes a fair driving schedule for all people in a carpool over $d$ days
- Order of the Rubik's cube group
- proving tautologically equivalent
- Prove that $5/2 < e < 3$?
- The Prime Polynomial : Generating Prime Numbers
- Proof that the Lebesgue measure is complete
- Why Vandermonde's determinant divides such determinant?