Selecting Balls From Urns With Partial Replacement Rules - Burden, Julian; Chakraborty, Chandramauli; Fang, Qizhou; Lin, Lisa; Malibari, Nasser; Matoush, Sammi; Milbank, Isaiah; Parekh, Zahan; Prado, Martín; Ren, Rachael; Rong, Qizhao; Sánchez Garza, Maximiliano; et al. (The PUMP Journal of Undergraduate Research, 2024)
Consider an urn with an initial state of R red balls and W white balls. Draw a ball from the urn, uniformly at random, and note its color. If the ball is white, do not replace it; if the ball is red, do replace it. Define this sampling rule to be "Preferential". We study the random variable X denoting the number of white balls drawn under the Preferential sampling rule for a sample size n. It is known that the expected number of X is bounded below by 3nW/(4N), and bounded above by nW/N. In this paper we improve the lower bound, give a heuristic for the best possible lower bound, and we explore some properties of a generalization of this sampling rule, we call "Super-Preferential", where the probability of retaining a white ball is w and the probability of retaining a red ball is r.