#include<bits/stdc++.h>
#define MAX 100005
using namespace std;
int n,p[MAX],q[MAX];
#define db long double
db f[1005][1005],g[1005][1005];
db h[MAX][3];
void sol(){
    cin>>n;
    for(int i=0;i<n;i++) cin>>p[i];
    for(int i=0;i<n;i++) cin>>q[i];
    h[n][0]=h[n][1]=h[n][2]=0;
    for(int i=n-1;~i;i--){
        for(int j=1;j<=p[i]&&i+j<=n;j++){
            h[i][0]=max(h[i][0],1-h[i+j][1]);
            h[i][1]=max(h[i][1],1-h[i+j][1]);
            h[i][2]=max(h[i][2],h[i+j][1]);
            if(j!=p[i]){
                db wn=f[p[i]-j][q[i]+q[i+j]];
                db ls=1-wn-g[p[i]-j][q[i]+q[i+j]];
                db rt=wn*h[i+j][0]+(1-wn-ls)*(1-h[i+j][1])+ls*(1-h[i+j][2]);
                //cout<<"i="<<i<<" "<<j<<" "<<wn<<" "<<ls<<endl;
                h[i][1]=max(h[i][1],rt);
            }
        }
        //cout<<h[i][0]<<" "<<h[i][1]<<" "<<h[i][2]<<endl;
    }
    cout<<setprecision(18)<<h[0][1];
}
int main(){
    ios::sync_with_stdio(0);
    cin.tie(0);
    int t=1;
    for(int i=1;i<=1000;i++){
        for(int j=1;j<=1000;j++){
            f[i][j]=f[i-1][j]*(i-1)/i;
            if(j<i) f[i][j]+=1.0/i;
            else f[i][j]+=1.0/(j+1);
            
            g[i][j]=g[i-1][j]*(i-1)/i;
            if(j>=i) g[i][j]+=1.0/(j+1)/i;
        }
    }
    while(t--) sol();
}