Transition Probability Matrix of a Particular Markov Chain

Question

I have come across a "MC that represents the success runs of binomial trials with success p".

So does the state of the MC represent the total number of successes for the trials?

The transition probability matrix is

$$ \begin{bmatrix}(1-p)&p&0&0&...\\(1-p)&0&p&0&...\\(1-p)&0&0&p&...\\...&...&...&...\end{bmatrix} $$

So e.g. state $n \rightarrow n+1 $ with probability p, but why does state $ n \rightarrow 0$ with probability (1-p)? Shouldn't it be state $ n \rightarrow n$ with probability (1-p) instead?

Is there a particular name for this Markov Chain?

Thank you!

Answer 1

This is the Markov chain for the runs of consecutive successes, not for the total number of successes.

Suppose that you are playing many rounds of a game, and you win each round with probability $p$. This Markov chain keeps track of your streak: how many wins have you had since the most recent loss? After each game, the streak changes as follows:

• If you win the game, your streak increases from $n$ to $n+1$.
• If you lose the game, your streak resets from $n$ to $0$.