Q.A transition matrix for weather is $P=\begin{pmatrix}0.8&0.2\\0.3&0.7\end{pmatrix}$ (rows/columns ordered Sunny, Rainy). If today's weather distribution is equally likely to be sunny or rainy, find tomorrow's probability distribution.

Multiplying today's (mixed) state vector by the transition matrix gives tomorrow's distribution.

Initial distribution

Equally likely today means $\pi_0=(0.5,0.5)$.

Computing tomorrow's distribution

$=(0.4+0.15,\ 0.1+0.35)=(0.55,0.45)$ …

Multiplying by $P$ column-wise as if $\pi_0$ were a column vector on the right of $P$ (i.e. computing $P\pi_0$) instead of the correct row-vec …