\(\newcommand{\N}{\mathbb{N}}\)

Alternating Coin Experiment

Description

A coin has probability of heads \(p \in (0, 1)\). There are \(n \in \N_+\) players who take turns tossing the coin in round-robin style: player 1 first, then player 2, continuing until player \(n\), then player 1 again, and so forth. The first player to toss heads wins the game. Let \(N\) denote the number of the first toss that results in heads; \(N\) has the geometric distribution on \(\N_+\) with parameter \(p\). Let \(R\) denote the number of complete tail rounds played (all players toss tails); \(R\) has the geometric distribution on \(\N\) with parameter \(1 - (1 - p)^n\). Let \(W\) denote the winner of the game; \(W\) has the truncated geometric distribution with parameter \(n\) and \(p\). The variables \(N\), \(R\), and \(W\) are related by \(N = n R + W\) and are recorded in the data table on each update. The distributions of the variables are described in the distribution graphs and tables. The coins show just the result of the last round, when the winning player is determined. The parameters \(n\) and \(p\) can be varied with the scrollbars.