Consider a simple model that progresses year-by-year. In year $i$, let $W_i$ = patient is well, $I_i$ = patient is ill, and $D_i$ = patient is dead. Transitions can be modeled as a set of conditional probabilities.
Let $L$ = number of years that the patient is well.
I have come up with the probability mass function of L to be $P(L=\ell)=(p)^{\ell-1}(1-p)$.
The next part of the problem is to consider the following scenario: 100 patients were all well in year 1 and the probability of being well in year 3 is estimated to be 80%. Assuming the transition probabilities are constant, what is $P(W_{i+1}|W_i)$?
My assumption was that I would plug in the information given from the studies. For example, for study 1: $P(L=3)=0.8=(p)^{2}(1-p)$ and then solve for $p$. However, according to Wolfram Alpha, this should result in $p=-6.88365$ which doesn't make sense for a probability value.
So first of all for solving your equation $p^2(1-p)\leqslant p(1-p)\leqslant1/4$, so this can't be solved. Assuming the transition probabilities are the same gives: $$ 0.8=p(W_3|W_1)=p(W_2|W_1)p(W_3|W_2)=p(W_2|W_1)^2. $$ This is saying that for a patient to be well in year 3, given that they were well in year 1, they can't get ill in year 2.
So, we have $$ p(W_{i+1}|W_i)=\frac{2}{\sqrt{5}}. $$
I'm a bit confused by the two studies, they seem to be word for word the same?