According to the Markov chain model, the Land of Oz has the following transition matrix for weather:

\[ P = \begin{pmatrix} 0.5 & 0.25 & 0.25 \\ 0.5 & 0 & 0.5 \\ 0.25 & 0.25 & 0.5 \end{pmatrix} \]

Given that it is raining today, we represent this initial condition with the state vector:

\[ w = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \]

To calculate the probability of the weather conditions after 1, 2, and 3 days, we use the following matrix multiplications:

\[ w^{(1)} = wP = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 0.5 & 0.25 & 0.25 \\ 0.5 & 0 & 0.5 \\ 0.25 & 0.25 & 0.5 \end{pmatrix} \]

\[ w^{(2)} = wP^2 = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \left(\begin{pmatrix} 0.5 & 0.25 & 0.25 \\ 0.5 & 0 & 0.5 \\ 0.25 & 0.25 & 0.5 \end{pmatrix}\right)^2 \]

\[ w^{(3)} = wP^3 = \begin{pmatrix} 1 & 0 & 0 \end{pmatrix} \left(\begin{pmatrix} 0.5 & 0.25 & 0.25 \\ 0.5 & 0 & 0.5 \\ 0.25 & 0.25 & 0.5 \end{pmatrix}\right)^3 \]

The results of these calculations give us the probability distribution for rain (R), nice weather (N), and snow (S) over the course of the next three days.