This app models the success-runs Markov chain corresponding to Bernoulli trials with probability of success \(p\). The state at any time is the length of the current run of successes. The limiting distribution is the geometric distribution on \(\N\) with parameter \(1 - p\).
The top graph shows the state space, with the current state colored red and the initial state colored blue. As the process runs, the chain moves from state to state. The time and the sate are recorded at each update in the data table. The limiting distribution and the proportion of time that the chain is in each state are shown in the distribution graph and distribution table. The parameter \(p\) and the initial state \(x_0\) can be varied with the input controls.