= Solution
Use <matrix multiplication> to evolve the <probability vector> by $x_{n+1}=Px_n$. With $x_0=(0.4,0.4,0.2)^{\mathsf T}$,
$$
x_1=\begin{pmatrix}0.24+0.04+0.02\\0.04+0.32+0.04\\0.12+0.04+0.14\end{pmatrix}=\begin{pmatrix}0.30\\0.40\\0.30\end{pmatrix},\qquad x_2=\begin{pmatrix}0.18+0.04+0.03\\0.03+0.32+0.06\\0.09+0.04+0.21\end{pmatrix}=\begin{pmatrix}0.25\\0.41\\0.34\end{pmatrix}.
$$
Hence, in Central, Headington, Botley order,
$$
\boxed{\text{day 1: }(30\%,40\%,30\%),\qquad\text{day 2: }(25\%,41\%,34\%).}
$$
Both <probability vectors> have nonnegative entries summing to one.
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