Remove["Global`*"];
H[x_, n_] := If[n==0,0,Sum[x[[i]],{i,1,n}]];
X={1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1};
α=2;
β=2;
n=0;
Manipulate[Plot[PDF[BetaDistribution[α+H[X,n],β+n-H[X,n]],x],{x,0,1}, PlotStyle->Thick,PlotRange->{{0,1},{0,10}}, Frame->{True,True,False,False}, FrameLabel->{"θ","P(θ)"}],{n,0,Length[X],1}]

Y=Table[RandomInteger[BernoulliDistribution[0.75]],{i,1,100}]
α=1;
β=1;
n=0;
Manipulate[Plot[PDF[BetaDistribution[α+H[Y,n],β+n-H[Y,n]],x],{x,0,1}, PlotRange->{{0,1},{0,10}},PlotStyle->Thick, Frame->{True,True,False,False}, FrameLabel->{"θ","P(θ)"}],{n,0,Length[Y],1}]