tirage1=function(p,l,n)
 { 
 suite=numeric(n) 
   for(j in 1:n) 
     { 
      k=sample(1:4,1)
      i=1 
         u=0
         vec=c()
      while(u==0&i<=k) 
          { 
            u=rbinom(1,1,p)
           vec=c(vec,u) 
           i=i+1 
          } 
       if(j<=l)cat(k,paste("    "),vec,"\n") 
       suite[j]=u
}  
prop=mean(suite)
res=list(prop=prop,proba=(1-(1-p)*(1-(1-p)^4)/(4*p)))
res
} 

tirage1(0.2,10,100000)

tirage2=function(p,n)
{
suite=logical(n)
for(j in 1:n)suite[j]=rgeom(1,p)+1<=sample(1:4,1)
 prop=mean(suite)                                            
 res=list(prop=prop,proba=(1-(1-p)*(1-(1-p)^4)/(4*p)))       
 res                                                         
 } 
 
 tirage2(0.2,100000)  
 
 tirage3=function(n,vq)
 {
 ddeux=numeric(n)
 for(i in 1:n)
 {
 ddeux[i]=sum((table(sample(1:4,200,replace=T))/200-0.25)^2) 
 }
 resu= quantile(ddeux,vq) 
 resu
 }
 
  tirage3(1000,c(0.1,0.25,0.50,0.75,0.9))
 
 
 
 
 
 
 
  
    
  
  
  
  tirage4=function(n1,n2,x)
  {
  vec=numeric(n1)
  for(j in 1:n1)vec[j]=tirage3(n2,x)  
   plot(vec,xlab="no essai",ylab="D9")
  abline(h=qchisq(x,3)/800)  
  abline(h=mean(vec),col="red") 
  title(" 200 essais, moyenne des essais et valeur théorique")
   }  
   
  tirage4(200,1000,0.9)
  
                                                   
  sondage=function(N,K,n,n0,k,alpha)
  {
  res=logical(n0)
  fb=numeric(n0)
  fh=numeric(n0)
  pop=c(rep(1,K),rep(0,N-K))
  prop=K/N
  qq=qnorm(1-(alpha/2))
  for(i in 1:n0)
  {
  ech=sample(pop,n)
  moy=mean(ech)
  s=sqrt(moy*(1-moy)/n)
  e=qq*s*sqrt((N-n)/(N-1))
  fb[i]=moy-e
  fh[i]=moy+e
  if(i <=k)cat(c(round(fb[i],4),round(fh[i],4)),"\n")
  }
  etend=range(fb,fh)
  plot(1:n0,fb,type="n",ylim=etend,xlab="no institut",ylab="Proportion observée")
  segments(1:n0,fb,1:n0,fh)
  propobs=mean((fb<prop)&(prop<fh))
  abline(h=prop)
  text((1+n0)/2,max(fh),paste("proportion de recouvrement de ",prop,"  = ",round(propobs,4)))
  title(paste("Fourchettes de ", n0," instituts de sondage"))
  }

  sondage(10000,4000,1000,200,10,0.05)