Consider the following balls-and-bin game. We start with one...

Consider the following balls-and-bin game. We start with one black ball and one white ball in a bin. We repeatedly do the following: choose one ball from the bin uniformly at random, and then put the ball back in the bin with another ball of the same color. We repeat until there are $n$ balls in the bin. Show that the number of white balls is equally likely to be any number between 1 and $n-1$.

Key Concepts

Pólya’s Urn Model

This is a classic model in probability theory in which, at each step, a ball is drawn at random from an urn and then replaced along with an additional ball of the same color. This reinforcement mechanism causes the probabilities to evolve in a dependent manner and serves as a foundation for studying self-reinforcing random processes.

Symmetry and Exchangeability

In the context of an urn that starts with one ball of each color, the process is symmetric since the initial conditions and replacement rules treat each color identically. This exchangeability means that the outcomes are invariant under any reordering of events, leading to conclusions about equal likelihood of certain final counts.

Martingale Properties

The proportion of balls of a particular color in the urn forms a martingale, meaning that its expected value remains unchanged over the course of the process. This property is significant in analyzing and proving the uniformity of the distribution of outcomes, as it provides a powerful tool for understanding the dynamics of the reinforcement process.

Combinatorial Probability

The proof that each possible count of white balls is equally likely involves combinatorial reasoning, where the number of distinct sequences of draws leading to a given configuration is considered. This approach reveals a uniform distribution among the possible outcomes by counting the equally probable paths in the process.

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