pmfY5 = 
        Table[{(x/5), (0.25^x) (0.75^(5 - x)) Binomial[5, x]}, {x, 0, 
            5}];
pmfY10 = 
        Table[{(x/10), (0.25^x) (0.75^(10 - x)) Binomial[10, x]}, {x, 0, 
            10}];
pmfY50 = 
        Table[{(x/50), (0.25^x) (0.75^(50 - x)) Binomial[50, x]}, {x, 0, 
            50}];
pmfY200 = 
        Table[{(x/200), (0.25^x) (0.75^(200 - x)) Binomial[200, x]}, {x, 0, 
            200}];

<< Graphics`MultipleListPlot`

{PlotSymbol[Triangle], PlotSymbol[Box]}

\(MultipleListPlot[pmfY5, pmfY10, pmfY50, pmfY200, 
      PlotJoined -> True, PlotRange -> {0,  .4}, 
      PlotLegend -> {" n=5", " n=10", " n=50", " n=200"}, LegendPosition -> {0.4, \(-0.27\)}, 
      SymbolShape -> {PlotSymbol[Box], PlotSymbol[Box], PlotSymbol[Box], 
          PlotSymbol[Box]}, 
      SymbolStyle -> {GrayLevel[ .2], GrayLevel[ .4], GrayLevel[ .7], 
          GrayLevel[ .9]}, Frame -> True, 
      FrameLabel -> {"x", \*""(Subscript[pmf, X])(x)""}, 
      RotateLabel -> False]\)

p = 0.25;
n = 5;
Table[{a, p - a/n, 
        Sum[(p^x) (((1 - p))^(n - x)) Binomial[n, x], {x, 
            Floor[p n] - 1, Ceiling[p n] + a}]}, {a, 0, 2}]

Ceiling[p n + 2]/n - p

p = 0.25;
n = 10;
Table[{a, p - a/n, 
        Sum[(p^x) (((1 - p))^(n - x)) Binomial[n, x], {x, 
            Floor[p n] - 2, Ceiling[p n] + a}]}, {a, 0, 5}]

Ceiling[p n + 3]/n - p

p = 0.25;
n = 50;
Table[{a, p - a/n, 
        Sum[(p^x) (((1 - p))^(n - x)) Binomial[n, x], {x, 
            Floor[p n] - a, Ceiling[p n] + a}]}, {a, 0, 10}]

Ceiling[p n + 7]/n - p

p = 0.25;
n = 200;
Table[{a, p - a/n, 
        Sum[(p^x) (((1 - p))^(n - x)) Binomial[n, x], {x, 
            Floor[p n] - a, Ceiling[p n] + a}]}, {a, 0, 20}]

Ceiling[p n + 16]/n - p

p = 0.005;
n = 200;
Table[{a, p - a/n, 
        Sum[(p^x) (((1 - p))^(n - x)) Binomial[n, x], {x, 
            Floor[p n] - 1, Ceiling[p n] + a}]}, {a, 0, 20}]

Ceiling[p n + 3]/n - p

ListPlot[{{0.005, 0.015}, {0.01, 0.02}, {0.05, 0.04}, {0.1, 
          0.055}, {0.25, 0.08}, {0.5, 0.09}, {0.75, 0.08}, {0.9, 
          0.055}, {0.95, 0.04}, {0.99, 0.02}, {0.995, 0.015}}, 
      PlotJoined -> True, PlotRange -> {0, 0.1}, 
      FrameLabel -> {"p", "a  "}, RotateLabel -> False, 
      Frame -> True, 
      Ticks -> {Table[i, {i, 0, 1, 0.1}], Automatic}]

ListPlot[{{200 0.995 0.005, 0.015}, {200 0.99 0.01, 
          0.02}, {200 0.95 0.05, 0.04}, {200 0.9 0.1, 
          0.055}, {200 0.75 0.25, 0.08}, {200 0.5 0.5, 
          0.09}, {200 0.25 0.75, 0.08}, {200 0.1 0.9, 
          0.055}, {200 0.05 0.95, 0.04}, {200 0.01 0.99, 
          0.02}, {200 0.005 0.995, 0.015}}, PlotJoined -> True, 
      PlotRange -> {0, 0.1}, Frame -> True, 
      FrameLabel -> {"VAR", "a  "}, 
      RotateLabel -> False]