(* Simple binomial example; probability of "success" is p=1/6; choose number of trials n=3 with the slider. What's plotted is the probability as a function of x number of "successes" *)

{Slider[Dynamic[n],{1,3,1}],Dynamic[ListPlot[Table[{x,PDF[BinomialDistribution[n,1/6],x]},{x,0,3}],Filling->Axis,PlotRange->All]],Dynamic[n]}

(* Example of binomial-- change p in the argument of BinomialDistribution to 0.5 to see symmetry. The slider controls the number of trials. *)

{Slider[Dynamic[n1],{1,10,1}],Dynamic[ListPlot[Table[{x,PDF[BinomialDistribution[n1,0.5],x]},{x,0,10}],Filling->Axis,PlotRange->All]],Dynamic[n1]}

(* Limit of binomial becomes Poisson for n large, p small.  The first plot is a binomial and the second plot is a Poisson with the same mean mu.  The slider controls n: they look more and more similar as n increases.  *)

{Slider[Dynamic[n2],{1,40,1}],Dynamic[ListPlot[Table[{x,PDF[BinomialDistribution[n2,0.1],x]},{x,0,20}],Filling->Axis,PlotRange->All]],Dynamic[ListPlot[Table[{x,PDF[PoissonDistribution[n2*0.1],x]},{x,0,20}],Filling->Axis,PlotRange->All]]
,Dynamic[n2]}

(* Gaussian limit for Poisson and Gaussian: mu=np large.  The first plot is a binomial and the second is a Poisson with the same mean.  The slider controls n.  They both look more and more Gaussian as n gets large. *)

{Slider[Dynamic[n3],{1,200,1}],Dynamic[ListPlot[Table[{x,PDF[BinomialDistribution[n3,0.1],x]},{x,0,30}],Filling->Axis,PlotRange->All]],Dynamic[ListPlot[Table[{x,PDF[PoissonDistribution[n3*0.1],x]},{x,0,30}],Filling->Axis,PlotRange->All]]
,Dynamic[n3]}

(*Integral of a Gaussian *)

Clear[mu]; Clear[sigma];

N[Integrate[Exp[-(x1-mu)^2/(2*sigma^2)]/(sigma*Sqrt[2*Pi]),{x1,mu-sigma,mu+sigma}]]

N[Integrate[Exp[-(x1-mu)^2/(2*sigma^2)]/(sigma*Sqrt[2*Pi]),{x1,mu-2*sigma,mu+2*sigma}]]

N[Integrate[Exp[-(x1-mu)^2/(2*sigma^2)]/(sigma*Sqrt[2*Pi]),{x1,mu-3*sigma,mu+3*sigma}]]

N[Integrate[Exp[-(x1-mu)^2/(2*sigma^2)]/(sigma*Sqrt[2*Pi]),{x1,mu-4*sigma,mu+4*sigma}]]