In a well-shuffled deck of cards, what is the probability that somewhere in the deck there are adjacent cards of the same rank? What is the average number of adjacent matches? What is the probability distribution for the number of matches? We answer these and related questions for both the standard $52$-card deck with four suits and $13$ ranks and for generalized decks with $k$ suits and $n$ ranks. We also determine the limiting distribution as $n$ goes to infinity with $k$ fixed.

We determine the exact probabilities of the different isomorphism classes of tournaments that result from random sets of three and four independent dice drawn from the balanced uniform model of 3-sided dice.

Rich ecosystems harbour thousands of species interacting in tangled networks encompassing predation, mutualism and competition. Such widespread biodiversity is puzzling, because in ecological models it is exceedingly improbable for large communities to stably coexist. One aspect rarely considered in these models, however, is that coexisting species in natural communities are a selected portion of a much larger pool, which has been pruned by population dynamics. Here we compute the distribution of the number of species that can coexist when we start from a pool of species interacting randomly, and show that even in this case we can observe rich, stable communities. Interestingly, our results show that, once stability conditions are met, network structure has very little influence on the level of biodiversity attained. Our results identify the main drivers responsible for widespread coexistence in natural communities, providing a baseline for determining which structural aspects of empirical communities promote or hinder coexistence.

We give a number theoretic proof of the integrality of certain BPS invariants of knots. The formulas for these numbers are sums involving binomial coefficients and the Möbius function. We also prove a conjecture about further divisibility properties of the invariants.

Summary We consider n-sided dice whose face values lie between 1 and n and whose faces sum to n(n + 1)/2. For two dice A and B, define A ≻ B if it is more likely for A to show a higher face than B. Suppose k such dice A1, …, Ak are randomly selected. We conjecture that the probability of ties goes to 0 as n grows. We conjecture and provide some supporting evidence that—contrary to intuition—each of the assignments of ≻ or ≺ to each pair is equally likely asymptotically. For a specific example, suppose we randomly select k dice A1, …, Ak and observe that A1 ≻ A2 ≻ … ≻ Ak. Then our conjecture asserts that the outcomes Ak ≻ A1 and A1 ≻ Ak both have probability approaching 1/2 as n → ∞.

We generalise the formula expressing the matrix trace of a given square matrix as the integral of the numerical values of $A$ over the Euclidean sphere to the unit spheres of finite-dimensional normed spaces that have a 1-symmetric basis. Our result is new even in the case of $\ell_p$-norms in $\mathbb{R}^N$ for $p\neq 2$.

In a guessing game, players guess the value of a random real number selected using some probability density function. The winner may be determined in various ways; for example, a winner can be a player whose guess is closest in magnitude to the target, or a winner can be a player coming closest without guessing higher than the target. We study optimal strategies for players in these games and determine some of them for two, three, and four players.

Proof. Write Ck as the sum of indicator random variables 1γ , for γ a k-cycle. This means that 1γ(π) is 1 if γ is a cycle of π and 0 otherwise. Then E(Ck) = ∑ γ E(1γ). To determine E(1γ) we count the number of permutations having γ as a cycle. That number is (n− k)!. Thus, E(1γ) = (n − k)!/n!. Now, the number of possible γ is n(n − 1) · · · (n − k + 1)/k, since a k-cycle is an ordered selection of k elements from n in which any of the k elements can be put first. Thus, E(Ck) = (n(n− 1) · · · (n− k + 1)/k)((n− k)!/n!) = 1/k.

SummaryA question in geometric probability about the location of the balls in a game of bocce leads to related questions about the probability that a system of linear equations has a positive solution and the probability that a random zero-sum game favors the row player. Under reasonable assumptions, we are able to find these probabilities.

The semigroup game is a two-person zero-sum game defined on a semigroup $${(S,\cdot)}$$ as follows: Players 1 and 2 choose elements $${x \in S}$$ and $${y \in S}$$, respectively, and player 1 receives a payoff f ( x y ) defined by a function f : S → [−1, 1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S , to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.

The original shipping strategy of FedEx is to fly all packages to a hub location during the afternoon and evening, sort them there, and then fly them to their destinations during the night for delivery the next day. This leads to interesting mathematical questions: Given a population represented by points in Euclidean space or on a sphere, what is the location of the point of the hub that minimizes the total distance to all the points? Is such a point unique? Then using census data from 2000 we examine how close the FedEx hub in Memphis is to the hub for the U.S. population.

Summary This paper surveys the fascinating mathematics of fair division, and provides a suite of examples using basic ideas from algebra, calculus, and probability which can be used to examine and test new and sometimes complex mathematical theories and claims involving fair division. Conversely, the classical cut-and-choose and moving-knife algorithms show it is often possible to express practical, yet clean, clear, and beautiful logical conclusions without using highly technical language.

We determine the probability that a random k-dimensional subspace of Euclidean n-space contains a positive vector.

Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie bracket of vectorfields on the manifold. The motion of a ball rolling on an oriented surface is the parallel transport of a similar connection on the trivial SO(3)-bundle over the surface. If the surface is a plane or a sphere, then the curvature of the connection is a scalar multiple of the Lie bracket in the Lie algebra of SO(3).