prob[p_,n1_,N_]:=Binomial[N,n1] (p^n1) (((1-p))^(N-n1))

prob[0.5,11,20]

ListPlot[Table[{k,PDF[BinomialDistribution[20,p],k]},{p,{0.5}},{k,0,20}],Filling->Axis,PlotStyle->PointSize[0.015],PlotRange->All,AxesOrigin->{0,0}]

ListPlot[Table[{k,PDF[BinomialDistribution[10000,p],k]},{p,{0.0002}},{k,0,20}],Filling->Axis,PlotStyle->PointSize[0.015],PlotRange->All,AxesOrigin->{0,0}]

ww[λ_,n_]:=((λ^n)/(n!)) (E^(-λ))

ListPlot[Table[ww[0.5 20,i],{i,0,20}],PlotStyle->PointSize[0.015]]