prob.tie<-function(n,p){
c(
  exact=sum(dbinom(0:n,n,p)^2),
  approx=1/sqrt(4*pi*n*p*(1-p))
 )
}

> prob.tie(10,.8)
    exact    approx 
0.2238044 0.2230155 

> prob.tie(100,.8)
     exact     approx 
0.07053579 0.07052370 

> prob.tie(1000,.8)
     exact     approx 
0.02230190 0.02230155 

> options(digits=12)

> prob.tie(1e4,.8)
           exact           approx 
0.00705238082422 0.00705236979435 

> prob.tie(1e5,.8)
           exact           approx 
0.00223015549369 0.00223015514519 

> prob.tie(1e6,.8)
            exact            approx 
0.000705236990454 0.000705236979435 

> prob.tie(1e7,.8)
            exact            approx 
0.000223015514868 0.000223015514519 

Accurate to 9 significant digits!