A machine breaks with probability $p$ and keeps working with probability $q=1-p$ on any day. When the machine is broken, it takes a random number of days $N$ to repair. N has a geometric distribution given by $P(N=k)=\beta(1-\beta)^{k-1}$. $beta$ is between $0$ and $1$. $X_n=1$ if the machine works on day $n$, $X_n=0$ if broken.
How would you write the transition matrix?
How do you find the transition probabilities of the Markov chain?
What is $P(N\geq2)$?
Since the number of days taken to repair the machine is geometrically distributed with parameter $\beta$, when the machine is broken, it will remain broken with probability $\beta$ and be repaired with probability $1-\beta$. (This follows from the memoryless property of the geometric distribution.) The transition matrix is then given by
$$ P = \begin{pmatrix}q&p\\1-\beta&\beta \end{pmatrix}. $$
The probability that the machine is broken for at least two days is given by
$$ 1 - (\beta + \beta(1-\beta)). $$