p1[E1_,E2_,E3_,T_]:=((Exp[-E1/T])/(Exp[-E1/T]+Exp[-E2/T]+Exp[-E3/T]));
p2[E1_,E2_,E3_,T_]:=((Exp[-E2/T])/(Exp[-E1/T]+Exp[-E2/T]+Exp[-E3/T]));
p3[E1_,E2_,E3_,T_]:=((Exp[-E3/T])/(Exp[-E1/T]+Exp[-E2/T]+Exp[-E3/T]));

e1=2;
e2=6;
e3=20;
Plot[{0.333333,p3[e1,e2,e3,t]},{t,0,50},PlotRange->{0,1.1}]

U[E1_,E2_,E3_,t_]:=e1 p1[E1,E2,E3,t]+e2 p2[E1,E2,E3,t]+e3 p3[E1,E2,E3,t]

e1=3;
e2=6;
e3=18;
Plot[{((e1+e2+e3)/3),U[e1,e2,e3,t]},{t,0,100},PlotRange->{0,All}]