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.

Solve $(a\ b)\begin{pmatrix}0.9&0.1\\0.3&0.7\end{pmatrix}=(a\ b)$ with $a+b=1$; this gives $a=0.75,\ b=0.25$.

In the TN HSC Class-12 Business Maths transition-probability topic, the equilibrium (steady-state) market shares form a state vector that is unchanged when multiplied by the transition matrix.

Step 1 — set up the equilibrium condition. Let the equilibrium shares be $a$ (brand A) and $b$ (brand B) with $a+b=1$. The transition matrix is

$T=\begin{pmatrix}0.9&0.1\\0.3&0.7\end{pmatrix}.$

Equilibrium requires $(a\ \ b)\,T=(a\ \ b)$.

Step 2 — form the equations.

$0.9a+0.3b=a,\qquad 0.1a+0.7b=b.$ …

Set the equilibrium state vector equal to itself after transition; solving $(a\;b)T=(a\;b)$ with $a+b=1$ gives $A=0.75,\ B=0.25$.

Let the equilibrium shares be $a$ (for $A$) and $b$ (for $B$) with $a+b=1$. The transition matrix is

$T=\begin{pmatrix}0.9 & 0.1\\ 0.3 & 0.7\end{pmatrix}.$

Equilibrium requires $(a\;\;b)T=(a\;\;b)$:

$0.9a+0.3b=a,\qquad 0.1a+0.7b=b.$ …