Needs["Graphics`Graphics`"]

Needs["Statistics`DiscreteDistributions`"]

Table[Random[Integer,{1,5}],{i,1,5}]

DisplayTogether[
Plot[0.5*(1-1/(10))^t,{t,0,500}],
Plot[0.5*(1-1/(100))^t,{t,0,500}],
Plot[0.5*(1-1/(1000))^t,{t,0,500}],
PlotRange->{Automatic,{0,0.5}}
]

Clear[na,i]
na[try_,n_,t_]:=na[try,n,t]=
	Random[BinomialDistribution[n,na[try,n,t-1]/n]];
na[try_,n_,0]=2;

For[j=1,j<21,For[k=1,k<50,na[j,20,k];k++];Print[na[j,20,50]];j++]

Clear[rn]
rn[try_,t_]:=rn[try,t]=Random[]

Clear[na,i]
na[try_,n_,t_]:=na[try,n,t]=
	(i=0;While[N[CDF[BinomialDistribution[n,na[try,n,t-1]/n],i]]<rn[try,t],i++];i);
na[try_,n_,0]=n/2;

DisplayTogether[
ListPlot[Table[{t,na[1,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[2,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[3,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[4,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[5,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[6,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[7,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[8,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[9,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[10,10,t]},{t,0,20}],PlotJoined->True],
PlotRange->{{0,20},{0,10}}]

DisplayTogether[
ListPlot[Table[{t,2*na[1,10,t]*(10-na[1,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[2,10,t]*(10-na[2,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[3,10,t]*(10-na[3,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[4,10,t]*(10-na[4,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[5,10,t]*(10-na[5,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[6,10,t]*(10-na[6,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[7,10,t]*(10-na[7,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[8,10,t]*(10-na[8,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[9,10,t]*(10-na[9,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[10,10,t]*(10-na[10,10,t])/100},{t,0,20}],PlotJoined->True],
PlotRange->{{0,20},{0,1}}]

DisplayTogether[
ListPlot[Table[{t,na[1,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[2,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[3,10,t]},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,na[4,10,t]},{t,0,20}],PlotJoined->True],
PlotRange->{{0,20},{0,10}}]

DisplayTogether[
ListPlot[Table[{t,2*na[1,10,t]*(10-na[1,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[2,10,t]*(10-na[2,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[3,10,t]*(10-na[3,10,t])/100},{t,0,20}],PlotJoined->True],
ListPlot[Table[{t,2*na[4,10,t]*(10-na[4,10,t])/100},{t,0,20}],PlotJoined->True],
PlotRange->{{0,20},{0,0.5}}]

DisplayTogether[
ListPlot[Table[{t,na[1,10,t]},{t,0,20}],PlotJoined->True],
PlotRange->{{0,10},{0,10}}]

DisplayTogether[
ListPlot[Table[{t,2*na[1,10,t]*(10-na[1,10,t])/100},{t,0,20}],PlotJoined->True],
PlotRange->{{0,10},{0,0.5}}]

Clear[t]
t={0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
While[Sum[t[[i]],{i,1,16}]>16 || Sum[t[[i]],{i,1,16}]<16,
t=Table[Random[PoissonDistribution[1]],{i,1,16}]]
t

Sum[t[[i]],{i,1,16}]