\documentclass[varwidth=true, border=2pt]{standalone}
\usepackage{units}
\usepackage{ifthen}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}
\tikzstyle{vertex}=[draw,black,fill=blue,circle,minimum size=10pt,inner sep=0pt]
\tikzstyle{edge}=[very thick]
\begin{tikzpicture}[scale=2.5]
    \node (a)[vertex,fill=gray!10,align=left] at (0,0) {$A~\nicefrac{1}{3}$\\$B~\nicefrac{1}{6}$\\$C~\nicefrac{1}{2}$};
    \node (b)[vertex,fill=gray!10,align=left] at (1,0)  {$A~\nicefrac{2}{5}$\\$B~\nicefrac{1}{5}$\\$C~\nicefrac{2}{5}$};

    \path[thick,->] (a)    edge node [anchor=center,above,sloped] {$\nicefrac{9}{10}$} (b);
    \path[thick,->] (a)    edge[loop above, looseness=5] node [anchor=center,above,sloped] {$\nicefrac{1}{10}$} (a);
    \path[thick,->] (b)    edge[loop above, looseness=5] node [anchor=center,above,sloped] {$1$} (b);
\end{tikzpicture}
\end{document}