sm={{(1-p)^3,(1-p)^2*p,(1-p)^2*p,(1-p)^2*p,(1-p)*p^2,(1-p)*p^2,(1-p)*p^2,p^3},{(1-p)^2,0,(1-p)*p,(1-p)*p,0,0,p^2,0},{(1-p)^2,(1-p)*p,0,(1-p)*p,0,p^2,0,0},{(1-p)^2,(1-p)*p,(1-p)*p,0,p^2,0,0,0},{0,0,1-p,0,0,0,p,0},{0,0,0,1-p,0,0,p,0},{0,0,0,1-p,0,p,0,0},{0,0,0,0,0,0,1,0}}/.p->0.1;

Graph[Map[StringTake[StringReplace[ToString[#],Space->""],{2,-2}]&,Subsets[{1,2,3}]]/.""->"None",DiscreteMarkovProcess[1,sm],VertexSize->{"Scaled",0.06},ImageSize->Medium]

𝒫=DiscreteMarkovProcess[1,sm];

Position[Subsets[{1,2,3}],{}|{2}|{3}]

Probability[x==1||x==3||x==4,x\[Distributed]StationaryDistribution[𝒫]]

Probability[x==1,x\[Distributed]StationaryDistribution[𝒫]]

Mean[FirstPassageTimeDistribution[DiscreteMarkovProcess[8,sm],1]]