There seem to be several misunderstandings and mistakes in your code.
r = {{0.9, 0.8333}, {0.8722, 0.8111}, {0.8528, 0.7667}, {0.8472, 0.7167}, {0.8583, 0.6667},
{0.8861, 0.6194}, {0.9167, 0.5806}, {0.9639, 0.5528}, {1.008, 0.5361}, {1.075, 0.5333}};
n = 10;
b[i_, n_, t_] := Binomial[n, i] t^i (1 - t)^(n - i);
B1[t_] := Sum[b[i, n, t] r[[i]], {i, 1, 10}];
R = ListPlot[r, Joined->True, PlotStyle->{Orange, Dashed}, PlotRange->Automatic, AspectRatio->Automatic];
Show[R, ParametricPlot[B1[t], {t, 0, 1}, PlotStyle->Cyan, PlotRange->Automatic, AspectRatio->Automatic],
PlotRange->Automatic]
Please compare my code with yours character by character looking for any tiny changes. When you find the changes then look in the documentation for Mathematica and try to understand why I would make such a change.
Even with my changes there are still problems with the first data points. I may have made mistakes or misunderstood. If you find that I have made a mistake and you can explain this then I will try to fix my error. Or it is possible that the data points need to be used in a different way.
