\[ \left[\begin{array}{cc} & P & S & U\\ P & .8 & .1 & .1 \\ S & .2 & .6 & .2 \\ U & .25 & .25 & .5 \end{array}\right] \]
\(P^2_{3,1}=[P_{3,1}*P_{1,1}]+[P_{3,2}*P_{2,1}]+[P_{3,3}*P_{3,1}]\)
(.25)*(.8) + (.25)*(.2) +(.5)*(.25)## [1] 0.375
\[ \left[\begin{array}{cc} & P & S & U\\ P & .8 & .1 & .1 \\ S & .2 & .6 & .2 \\ U & .25 & .25 & .5 \end{array}\right] \]
\(P^2_{3,1}=[P_{3,1}*P_{1,1}]+[P_{3,2}*P_{2,1}]+[P_{3,3}*P_{3,1}]\)
(.25)*(.8) + (.25)*(.2) +(.5)*(.25)## [1] 0.375