**83**

votes

**10**answers

9k views

### Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain.
I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...

**72**

votes

**24**answers

19k views

### What is convolution intuitively?

If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...

**56**

votes

**9**answers

9k views

### Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ ...

**55**

votes

**17**answers

34k views

### Google question: In a country in which people only want boys [closed]

Hi all!
Google published recently questions that are asked to candidates on interviews. One of them caused very very hot debates in our company and we're unsure where the truth is. The question is:
...

**54**

votes

**20**answers

7k views

### Probabilistic Proofs of Analytic Facts

What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...

**52**

votes

**5**answers

6k views

### Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?

I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...

**51**

votes

**3**answers

2k views

### Perron number distribution

A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...

**50**

votes

**12**answers

13k views

### What are the big problems in probability theory?

Most branches of mathematics have big, sexy famous open problems. Number theory has the Riemann hypothesis and the Langlands program, among many others. Geometry had the PoincarĂ© conjecture for a long ...

**48**

votes

**9**answers

6k views

### How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and ...

**47**

votes

**3**answers

2k views

### Flipping coins on a budget

A coin is flipped $n$ times and you win if it comes up heads at least $k$ times. The coin is unusual in that you're allowed to pick the probability $p_i$ that it comes up heads on the $i$th flip, ...

**45**

votes

**1**answer

4k views

### The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...

**44**

votes

**9**answers

6k views

### Does War have infinite expected length?

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers.
The ...

**43**

votes

**0**answers

2k views

### Alternating colors on a line: infinitely often or converge?

Suppose we have intervals of alternating color on $\mathbb{R}$ (say, red / blue / red / blue / ...). All intervals have independent length, with all red intervals distributed as $\mathbb{P}_{R}$, all ...

**40**

votes

**4**answers

2k views

### Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...

**38**

votes

**2**answers

1k views

### Topple height of randomly stacked bricks

What is the expected height of a stack of unit-length bricks, each one
stacked on the previous with a uniformly random shift within $\pm \delta$?
The stack topples if the center of gravity of the top ...

**37**

votes

**1**answer

4k views

### The human body's random number generator

I remember learning in microbiology that the human body generates antibodies using a random process so that an enormous variety of antibodies can be produced with a simple genetic code.
Now that I'm ...

**36**

votes

**8**answers

3k views

### A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it.
I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which
has the ...

**36**

votes

**7**answers

3k views

### Conway's game of life for random initial position

What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). Specifically ...

**35**

votes

**1**answer

2k views

### Rolling a random walk on a sphere

A ball rolls down an inclined plane, encountering horizontal obstacles, at which it
rolls left/right with equal probability. There are regularly spaced staggered gaps that let the ball
roll down to ...

**35**

votes

**4**answers

3k views

### Connectivity of the ErdÅ‘sâ€“Rényi random graph

It is well-known that if $\omega=\omega(n)$ is any function such that $\omega \to \infty$ as $n \to \infty$, and if $p \ge (\log{n}+\omega) / n$ then the ErdÅ‘sâ€“RĂ©nyi random graph $G(n,p)$ is ...

**34**

votes

**4**answers

2k views

### Polynomials on the Unit Circle

I asked this question in math.stackexchange but I didn't have much luck. It might be more appropiate for this forum. Let $z_1,z_2,â€¦,z_n$ be i.i.d random points on the unit circle ($|z_i|=1$) with ...

**34**

votes

**1**answer

2k views

### Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**33**

votes

**3**answers

2k views

### the following inequality is trueï¼Œbut I can't prove it

The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify ...

**32**

votes

**6**answers

962 views

### Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$,
one by one, and create a random jumble of disks above the $x$-axis.
When a falling disk hits another, it stops and sticks there.
Otherwise, ...

**32**

votes

**8**answers

1k views

### How to quantify noncommutativity?

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...

**32**

votes

**7**answers

2k views

### The shortest path in first passage percolation

Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and differ by one in the other.)
...

**32**

votes

**1**answer

1k views

### Modeling question: how often does “the world's oldest person” die?

This story yesterday (no need to follow the link to understand the question!)
http://www.cnn.com/2011/US/02/01/texas.oldest.person.dies/index.html?hpt=T2
reminds me that I've often wondered about ...

**30**

votes

**5**answers

4k views

### Heuristically false conjectures

I was very surprised when I first encountered the Mertens conjecture. Define
$$ M(n) = \sum_{k=1}^n \mu(k) $$
The Mertens conjecture was that $|M(n)| < \sqrt{n}$ for $n>1$, in contrast to the ...

**30**

votes

**1**answer

3k views

### 100â‚¬ bounty ended: Do invariant measures maximize the integral?

Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...

**29**

votes

**14**answers

23k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...

**29**

votes

**6**answers

4k views

### Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...

**29**

votes

**1**answer

3k views

### Anti-concentration bound for permanents of Gaussian matrices?

In a recent paper with Alex Arkhipov on "The Computational Complexity of Linear Optics," we needed to assume a reasonable-sounding probabilistic conjecture: namely, that the permanent of a matrix of ...

**29**

votes

**1**answer

1k views

### What is the sandpile torsor?

Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...

**29**

votes

**4**answers

2k views

### Geometric / physical / probabilistic interpretations of Riemann zeta(n>1)?

What are some physical, geometric, or probabilistic interpretations of the values of the Riemann zeta function at the positive integers greater than one?
I've found some examples:
1) In MO-Q111339 ...

**29**

votes

**1**answer

5k views

### “psi-epistemic theories” in 3 or more dimensions

In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for ...

**28**

votes

**1**answer

2k views

### An Entropy Inequality

Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the ...

**27**

votes

**3**answers

2k views

### Perron-Frobenius “inverse eigenvalue problem”

The Perron-Frobenius theorem says that the largest eigenvalue of a positive real matrix (all entries positive) is real. Moreover, that eigenvalue has a positive eigenvector, and it is the only ...

**27**

votes

**4**answers

1k views

### Probability of zero in a random matrix

Let $M(n,k)$ be the set of $n\times n$ matrices of nonnegative integers such that every row and every column sums to $k$. Let $P(n,k)$ be the fraction of such matrices which have no zero entries, ...

**27**

votes

**5**answers

2k views

### You pass X people and Y people pass you: how relatively fast are you?

This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...

**26**

votes

**3**answers

952 views

### Random knot on six vertices

This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...

**25**

votes

**2**answers

1k views

### “Are you more intelligent than the average of those who are more intelligent than the average?”

I'm sure that many MO users would answer "Oh, yes, I'm more intelligent than the average intelligence of the population that has an intelligence greater than the (absolute) average". And someone, less ...

**25**

votes

**3**answers

2k views

### Expectation of a random sum

Let $X_1, X_2, X_3,\dots$ be an i.i.d. sequence of random variables with finite mean. Write $S_n=X_1+X_2+\dots+X_n$.
Let $N$ be a non-negative integer-valued random variable with finite mean. $N$ may ...

**25**

votes

**3**answers

537 views

### What is the probability two random maps on n symbols commute?

It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...

**24**

votes

**1**answer

1k views

### A puzzle about finding three points $(x,y)$, $(x,z)$ and $(y,z)$ in a subset of a square.

I was asked (by myself) to give a proof of the following seemingly simple geometric statement, but after thinking a little I now suspect it could be less elementary than I thought (or am I being ...

**24**

votes

**5**answers

2k views

### Random walk: police catching the thief

I posted this problem on stackexchange.com,but haven't get a satifactory answer.
This is a problem about the meeting time of several independent random walks on the lattice $\mathbb{Z}^1$:
Suppose ...

**23**

votes

**2**answers

922 views

### The probability for a symmetric matrix to be positive definite

Let me give a reasonable model for the question in the title. In ${\rm Sym}_n({\mathbb R})$, the positive definite matrices form a convex cone $S_n^+$. The probability I have in mind is the ratio ...

**23**

votes

**2**answers

576 views

### What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...

**22**

votes

**11**answers

6k views

### Is pi a good random number generator?

Part of what I do is study typical behavior of large combinatorial structures by looking at pseudorandom instances. But many commercially available pseudorandom number generators have known defects, ...

**22**

votes

**7**answers

3k views

### What's the use of a complete measure?

A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...

**22**

votes

**3**answers

1k views

### Some models for random graphs that I am curious about

G(n,p)
We are familiar with the standard notion of random graphs where you fixed the number n of vertices and choose every edge to belong to the graph with probability 1/2 (or p) independently. This ...