Pepys' Problem

Statement: Toss 6n fair die; what is the probability of at least n sixes?

ncr[x_,y_] := x! / (y! (x-y)!)

probn[n_] :=1 - (Sum[ncr[6n,k] 5^(6n-k), {k,0,n-1}] / 6^(6n));
probn[1.]

tempplot = {};
fiftypercent = {};
For[n=1, n≤1000, n++, 
{
fiftypercent = AppendTo[fiftypercent, {n,.5}];
tempplot = AppendTo[tempplot, {n,1.0 probn[n]}]
}];
ListPlot[{tempplot,fiftypercent}]

tempplot = {};
For[n=1, n≤1000, n++, 
{
fiftypercent = AppendTo[fiftypercent, {n,.5}];
tempplot = AppendTo[tempplot, {n,1.0 probn[n]}]
}];
ListPlot[{tempplot,fiftypercent}]

probn[10000]

randomtrial[n_,iter_] := Module[{},
results = {};
For[i = 1, i ≤ iter, i++,
{
num6 = 0;
For[k = 1, k ≤ 6 n, k++, If[Random[] ≤ 1/6, num6 = num6+1]];
results = AppendTo[results, (num6 - n) / Sqrt[(1/6)(5/6)(6 n)]];
}];
Print[Histogram[results, {-4,4,.25},  "Probability", PlotLabel ->{"n=", n, "Histogram Plot: (#6s - expected)/StDev"}]];
];

randomtrial[300,1000]

For[ndo = 10, ndo ≤ 200, ndo = ndo + 10,
randomtrial[ndo,8000]]