In a Given Town, on a Given Day, Citizens Can $$\left( \begin{array} { l l l } .23 & .21 & .56 \\.15 & .41 & .44 \\.22 & .08 & .70\end{array} \right) .$$

Short Answer

In a given town, on a given day, citizens can either go to the library (activity 1), go to the zoo (activity 2), or stay home (activity 3). Let Xₙ=i if on the nᵗʰ day of the year, the Shuzoku family does activity i. Suppose {Xₙ | n=1,…} is a Markov chain with transition probability matrix $$\left( \begin{array} { l l l } .23 & .21 & .56 \\.15 & .41 & .44 \\.22 & .08 & .70\end{array} \right) .$$ We know the Shuzoku family went to the zoo on day 1. What is the probability they did not go to the zoo the next two days?