The elliptical case of an odds inversion problem

Hilmer, Kieran; Jin, Angela; Lycan, Ron; Ponomarenko, Vadim

doi:10.2140/involve.2023.16.431

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The elliptical case of an odds inversion problem

Kieran Hilmer, Angela Jin, Ron Lycan and Vadim Ponomarenko

Vol. 16 (2023), No. 3, 431–452

DOI: 10.2140/involve.2023.16.431

Abstract

A recent paper by R. Moniot investigates the problem of, given a probability $\frac{p}{q}$ , finding a number of red and blue balls such that, when drawing two balls without replacement, the probability of drawing different colored balls is $\frac{p}{q}$ . In this paper we deepen our understanding of the case where $\frac{p}{q} > \frac{1}{2}$ by finding bounds of the number of solutions for a given probability $\frac{m}{2 m - 1}$ with $m \in ℕ$ and characterize “families” of probabilities that are guaranteed to have more than two solutions. We also estimate the number of achievable probabilities in the ranges $[\frac{m}{2 m - 1}, 1]$ and $(\frac{m + 1}{2 m + 1}, \frac{m}{2 m - 1})$ . Finally, we show that the “recycling recurrence” only exists for $x_{1} = n^{2} - n$ , $y_{1} = n^{2}$ , and $y_{2} = n^{2} + n$ for $n \in ℕ$ .

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Keywords

odds inversion, recycling recurrence, probability, balls, urns

Mathematical Subject Classification

Primary: 11A99, 11D09

Secondary: 11Z05

Milestones

Received: 2 January 2022

Revised: 10 June 2022

Accepted: 13 June 2022

Published: 10 August 2023

Communicated by Scott T. Chapman

Authors

Kieran Hilmer
San Diego State University San Diego, CA United States

Angela Jin
Torrey Pines High School San Diego, CA United States

Ron Lycan
San Diego State University San Diego, CA United States

Vadim Ponomarenko
Department of Mathematics and Statistics San Diego State University San Diego, CA United States

Vol. 16, No. 3, 2023