I finish to do an exercise but I am not quite sure how to resolve the last question. Can you help me, please? Thanks in advance.
Exercise: Sylvie is in good health (state 1) or she is sick (state 2). In 98% of cases, when she is in good health on a given day, she will still be healthy the next day. In addition, in 40% of cases, when she is sick on a given day, she will regain her health the next day. Let T be the transition matrix, where t(ij) represents the probability that Sylvie will move from state j to state i.
(a)If Sylvie is sick today, use matrix multiplication to determine the probability that it will still be in two days. (b)In the long term, what percentage of days will Sylvie be healthy?
What I've done: The transition matrix:$ $ \begin{matrix} 0.98 & 0.4 \\ 0.02 & 0.6 \\ \end{matrix}
Let P(0) be the vector of the initial state :$ $ \begin{matrix} 0 \\ 1 \\ \end{matrix}
Let's calculate P(2).
P(2) = T.T.P(0) which make:$ $ \begin{matrix} 0.632\\ 0.368 \\ \end{matrix}
So there is a probability of 36.8% that Sylvie will still be sick in two days.