The following problem arises in the analysis of Bloom filters. Consider $m$ bins and $N=nk$ balls placed uniformly at random into the bins. A query chooses $k$ bins uniformly at …

Inspired by this question, I would like to determine the probability that a random knot of 6 unit sticks is a trefoil. This naturally leads to the following question: Is there a …

This question is inspired by Joseph O'Rourke's beautiful answer to my previous question. Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns hav …

Consider a Erdős–Rényi graph on $n$ nodes, say $1,2,\ldots,n$, ($n\geq 3$) such that the probability of edge between any two nodes is $c/n$. I wish to know if there is a result tha …

I have four solutions which are termed: A1, A2, A3, A4. These are actually the results of a searching algorithm. I know that A1 is the best solution, A2 is next to A1, A3 is next t …

The same colleague as in http://mathoverflow.net/questions/130489/another-colored-balls-puzzle asked me the following variant which she called "part II". Imagine you have $n$ ball …

Consider an experiment consisting of a repeated trial with two random Bernoulli (=binary) variables, A and B. Each trial consists of multiple outcomes for both A and B. Each trial …

[A duplicate thread can also be found at http://stats.stackexchange.com/questions/59450/finding-conditions-on-unspecified-cdf-that-permit-a-solution-to-an-equation ] Let $F(\alpha …

I was sent this puzzle and wondered if it is known or if its origin is known? (I see colored ball puzzles are also in vogue.) Consider a bag with $n$ red balls and $n$ blue balls. …

There is probably terminology for this, and I apologize that I don't know it, and part of my question is what the standard terminology for the concepts I'm giving is. This is a pr …

This is a puzzle a colleague asked me recently. Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have diff …

My question may be a bit lousy. Suppose we have a set of statistical variables X1, X2, .. Xn, and we have N independent samples. We can compute the covariance matrix of {X}. My que …

$\newcommand{\bsV}{\boldsymbol{V}}$ $\newcommand{\bsE}{\boldsymbol{E}}$ $\newcommand{\bR}{\mathbb{R}}$ Suppose that $\bsV$ is an $N$-dimensional real Euclidean space. Denote by …

I'm interested in some instances of the following problem. Let $n \geq 2$, and suppose we draw $k \geq 2$ vectors $v_1, \dots, v_k$ uniformly at random from the $n$-dimensional …

Assume the $1 \times N$ vector $\mathbf X = [X_1, X_2, \ldots , X_N]$ contains i.i.d. normal samples such that $\mathbf X$ has a multivariate normal distribution. Now assume anot …