Got given this question:
In a football tournament a player never gets 2 yellow cards (YCs) in the same match but can get issued with one YC (with probability p) independently of other matches. The YC stays with him through any subsequent matches in the tournament, until he gets a 2nd YC when he is suspended for one match and then his YC count reverts to zero. Let X be the random variable denoting the number of YCs a player currently holds. Set this up as a Markov chain and find the long term proportion of matches that the player misses through suspension.
How would I draw this transition diagram? Do I need to represent the suspended games in the diagram? And how would I go about finding the long term proportion of suspended matches?
Thank you!
Updating the answer, as per David's comments.
Assume the following state space, which represents the state a player is in before the game.
State 0: No yellow cards
State 1: 1 yellow card
State 2: Suspended
Then the transition matrix can be written as follows.
\begin{equation} P = \begin{bmatrix} 1-p & p & 0 \\ 0 & 1-p & p \\ 1 & 0 & 0 \end{bmatrix} \end{equation}
This assumes that if a player picks up a second YC during a match, he isn't technically suspended until the next match.
If the long run proportion of being in state $j$ is denoted by $\pi_j$, we can find the long run proportions by solving the following system of equations.
\begin{align} 1 &= \pi_0 + \pi_1 + \pi_2 \\ \pi_0 &= (1-p)\pi_0 + \pi_2 \\ \pi_1 &= p\pi_0 + (1-p)\pi_1 \\ \pi_2 &= p\pi_1 \end{align}