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n = 60000; q = 0.495; NSim = 10^5; sigma = Sqrt[4 q (1 - q)];
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sims = Table[Total[RandomChoice[{q, 1 - q} -> {1, -1}, n]], {NSim}];
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distr = HistogramDistribution[sims];
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Show[
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 DiscretePlot[PDF[distr, x], {x, -1200, 600}],
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 Plot[PDF[NormalDistribution[-600, Sqrt[n]*sigma], x], {x, -1200, 
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   600}] 
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 ]
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 ]
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Max[Table[Total[RandomChoice[{q, 1 - q} -> {1, -1}, n]], {10^5}]]