设\(f[i]\)为从\(i\)到\(r+1\)且不走出区间的概率
\(f[i]=p[i]f[i+1]+(1-p[i])f[i-1]\)
\(f[i]-f[i-1]=p[i](f[i+1]-f[i-1])\)
\(f[r+1]=1,f[l-1]=0\)
\(g[i]=f[i]-f[i-1]\)
\(g[i]=p[i](g[i+1]+g[i])\)
\(g[i+1]=\frac{1-p[i]}{p[i]} g[i]\)
\(\sum_{i=l}^{r+1} g[i]=f[r+1]-f[l-1]=1\)
\(lis[l][r]=\sum_{i=l}^{r} \prod_{j=l}^i \frac{1-p[j]}{p[j]}\)
\(f[l]=g[l]=\frac{1}{lis[l][r]+1}\)
线段树维护前缀积、前缀和即可
#include"cstdio"
#include"cstring"
#include"iostream"
#include"algorithm"
using namespace std;
const int MAXN=1<<17;
int n,m;
int ik[MAXN];
double sum[2][MAXN<<1];
struct rpg{double a,b;};
int read()
{
int x=0;char ch=getchar();
while(ch<'0'||'9'<ch) ch=getchar();
while('0'<=ch&&ch<='9') x=(x<<3)+(x<<1)+(ch^48),ch=getchar();
return x;
}
void build(int k,int l,int r)
{
if(l==r){ik[l]=k;return;}
int i=k<<1,mid=l+r>>1;
build(i,l,mid),build(i|1,mid+1,r);
return;
}
void cchg(int k,double v)
{
sum[0][k]=sum[1][k]=v;k>>=1;
while(k){
int i=k<<1;
sum[0][k]=sum[0][i]*sum[0][i|1];
sum[1][k]=sum[1][i]+sum[0][i]*sum[1][i|1];
k>>=1;
}return;
}
rpg cask(int k,int l,int r,int le,int ri)
{
if(le<=l&&r<=ri) return (rpg){sum[0][k],sum[1][k]};
int i=k<<1,mid=l+r>>1;
if(le<=mid&&mid<ri){
rpg sum=cask(i,l,mid,le,ri),tmp=cask(i|1,mid+1,r,le,ri);
sum.b+=sum.a*tmp.b;
sum.a*=tmp.a;
return sum;
}if(le<=mid) return cask(i,l,mid,le,ri);
return cask(i|1,mid+1,r,le,ri);
}
int main()
{
n=read();m=read();
build(1,1,n);
for(int i=1;i<=n;++i){
int a=read(),b=read();
double f=1.0*a/b;
cchg(ik[i],(1.0-f)/f);
}while(m--){
int p=read();
if(p==1){int k=read(),a=read(),b=read();double f=1.0*a/b;cchg(ik[k],(1.0-f)/f);}
else{int l=read(),r=read();printf("%.10lf\n",1.0/(cask(1,1,n,l,r).b+1));}
}return 0;
}