Answer:
The probability of the system being down in the next hour of operation is 0.3.
Step-by-step explanation:
We have a transition matrix from one period to the next (one hour) that can be written as:
[tex]T=\left[\begin{array}{ccc}&R&D\\R&0.7&0.3\\D&0.2&0.8\end{array}\right][/tex]
We can represent the state that system is initially running with the vector:
[tex]S_0=\left[\begin{array}{cc}1&0\end{array}\right][/tex]
The probabilties of the states in the next period can be calculated using the matrix product of the actual state and the transition matrix:
[tex]S_1=S_0\cdot T[/tex]
That is:
[tex]S_1=S_0\cdot T= \left[\begin{array}{cc}1&0\end{array}\right]\cdot \left[\begin{array}{cc}0.7&0.3\\0.2&0.8\end{array}\right]= \left[\begin{array}{cc}0.7&0.3\end{array}\right][/tex]
With the inital state as running, we have a probabilty of 0.7 that the system will be running in the next hour and a probability of 0.3 that it will be down.
