= Solution
The fitted <continuous-time multi-state model> allows only adjacent fatigue-state jumps. Its four <transition intensities>, in reciprocal years, are shown here:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/iii/paper-38-fatigue-transitions.png]
{title=Estimated fatigue transition intensities: mild to moderate 0.173, moderate to mild 0.4684, moderate to severe 0.3835, and severe to moderate 0.3655 per year}
In state $2$, the total exit intensity is $\lambda_2=q_{21}+q_{23}=0.4684+0.3835=0.8519$. By the <holding time> law for a <continuous-time Markov chain>, the time until the next jump has an <exponential distribution> with this rate. Therefore
$$
\boxed{\widehat{\mathbb E}T_2=\frac1{0.8519}=1.17385\text{ years}.}
$$
The printed interval for $q_{22}=-\lambda_2$ gives the interval $(0.7219,1.005)$ for $\lambda_2$. Taking reciprocals reverses the endpoints, so the transformed $95\%$ <confidence interval> for the mean <holding time> is
$$
\boxed{\left(\frac1{1.005},\frac1{0.7219}\right)=(0.9950,1.3852)\text{ years}.}
$$
For the next jump's destination, competing outgoing intensities give
$$
\boxed{\Pr\{2\to1\mid\text{a jump out of }2\}=\frac{q_{21}}{q_{21}+q_{23}}=\frac{0.4684}{0.8519}=0.54983.}
$$
Indeed the density of a first jump to $1$ at time $t$ is $q_{21}e^{-\lambda_2t}$; integrating gives the ratio above. This is a conditional next-jump probability, not a <transition probability> of being mild at a fixed later visit.
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