The Probability Distribution for Draws Until First Success Without Replacement

TL;DR

This paper analyzes the distribution of the number of draws without replacement until a good object is selected, computing its variance and demonstrating convergence to the geometric distribution.

Contribution

It provides a detailed analysis of the distribution's variance and proves its convergence to the geometric distribution, extending standard textbook exercises.

Findings

Variance of the distribution is explicitly computed.

Distribution converges to the geometric distribution.

Provides theoretical insights into the urn problem.

Abstract

We consider the urn setting with two different objects, ``good'' and ``bad'', and analyze the number of draws without replacement until a good object is picked. Although the expected number of draws for this setting is a standard textbook exercise, we compute the variance, and show that this distribution converges to the geometric distribution.