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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
Abstract

A recent paper by R. Moniot investigates the problem of, given a probability 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 p q. In this paper we deepen our understanding of the case where p q > 1 2 by finding bounds of the number of solutions for a given probability m 2m1 with m 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 [ m 2m1,1] and ( m+1 2m+1, m 2m1). Finally, we show that the “recycling recurrence” only exists for x1 = n2 n, y1 = n2, and y2 = n2 + n for n .

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