Suppose each day there is a $0.2$ probability will rain in the morning. $P(\text{rains afternoon}|\text{rain morning})=0.6$ and $P(\text{rains afternoon}|\text{not rain morning})=0.3$. Suppose John would go to the office in the morning and leave in the afternoon and he totally get 3 umbrellas. Let $X_n$ be the number of umbrellas storing in his office at $nth$ night. Whats the proportion of time he has $0$ umbrella in his office at night?
I tried to make the transition probability matrix and find the stationary distribution at state $0$ and found that $\pi_0=0.19$ but seems not consistent with the answer but i don't know what is the mistake. Any hints or solution are welcome.
$P_{transition} =\begin{pmatrix}0.92&0.08&0&0\\0.24&0.68&0.08&0\\0&0.24&0.68&0.08\\0&0&0.24&0.76\end{pmatrix}$ so $P-I=\begin{pmatrix}-0.08&0.08&0&0\\0.24&-0.32&0.08&0\\0&0.24&-0.32&0.08\\0&0&0.24&-0.24\end{pmatrix}$