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Physical Sciences and Mathematics Commons™
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- Probability (2)
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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis
Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis
Rose-Hulman Undergraduate Mathematics Journal
Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Divisibility Probabilities For Products Of Randomly Chosen Integers, Noah Y. Fine
Rose-Hulman Undergraduate Mathematics Journal
We find a formula for the probability that the product of n positive integers, chosen at random, is divisible by some integer d. We do this via an inductive application of the Chinese Remainder Theorem, generating functions, and several other combinatorial arguments. Additionally, we apply this formula to find a unique, but slow, probabilistic primality test.
A Model For The Multi-Virus Contact Process, Xu Huang
A Model For The Multi-Virus Contact Process, Xu Huang
Rose-Hulman Undergraduate Mathematics Journal
We study one specific version of the contact process on a graph. Here, we allow multiple infections carried by the nodes and include a probability of removing nodes in a graph. The removal probability is purely determined by the number of infections the node carries at the moment when it gets another infection. In this paper, we show that on any finite graph, any positive value of infection rate $\lambda$ will result in the death of the process almost surely. In the case of $d$-regular infinite trees, We also give a lower bound on the infection rate in order for …
Number Of Regions Created By Random Chords In The Circle, Shi Feng
Number Of Regions Created By Random Chords In The Circle, Shi Feng
Rose-Hulman Undergraduate Mathematics Journal
In this paper we discuss the number of regions in a unit circle after drawing n i.i.d. random chords in the circle according to a particular family of distribution. We find that as n goes to infinity, the distribution of the number of regions, properly shifted and scaled, converges to the standard normal distribution and the error can be bounded by Stein's method for proving Central Limit Theorem.
A Characterization Of Complex-Valued Random Variables With Rotationally-Invariant Moments, Michael L. Maiello
A Characterization Of Complex-Valued Random Variables With Rotationally-Invariant Moments, Michael L. Maiello
Rose-Hulman Undergraduate Mathematics Journal
A complex-valued random variable Z is rotationally invariant if the moments of Z are the same as the moments of W=e^{i*theta}Z. In the first part of the article, we characterize such random variables, in terms of "vanishing unbalanced moments," moment and cumulant generating functions, and polar decomposition. In the second part, we consider random variables whose moments are not necessarily finite, but which have a density. In this setting, we prove two characterizations that are equivalent to rotational invariance, one involving polar decomposition, and the other involving entropy. If a random variable has both a density and moments which determine …
Strong Recovery In Group Synchronization, Bradley Stich
Strong Recovery In Group Synchronization, Bradley Stich
Rose-Hulman Undergraduate Mathematics Journal
The group synchronization problem is to estimate unknown group elements at the vertices of a graph when given a set of possibly noisy observations of group differences at the edges. We consider the group synchronization problem on finite graphs with size tending to infinity, and we focus on the question of whether the true edge differences can be exactly recovered from the observations (i.e., strong recovery). We prove two main results, one positive and one negative. In the positive direction, we prove that for a sequence of synchronization problems containing the complete digraph along with a relatively well behaved prior …
Probabilities Involving Standard Trirectangular Tetrahedral Dice Rolls, Rulon Olmstead, Doneliezer Baize
Probabilities Involving Standard Trirectangular Tetrahedral Dice Rolls, Rulon Olmstead, Doneliezer Baize
Rose-Hulman Undergraduate Mathematics Journal
The goal is to be able to calculate probabilities involving irregular shaped dice rolls. Here it is attempted to model the probabilities of rolling standard tri-rectangular tetrahedral dice on a hard surface, such as a table top. The vertices and edges of a tetrahedron were projected onto the surface of a sphere centered at the center of mass of the tetrahedron. By calculating the surface areas bounded by the resultant geodesics, baseline probabilities were achieved. Using a 3D printer, dice were constructed of uniform density and the results of rolling them were recorded. After calculating the corresponding confidence intervals, the …