Algorithm

C. First Digit Law
time limit per test
2 seconds
memory limit per test
256 megabytes
problem link- https://codeforces.com/contest/54/problem/C
input
standard input
output
standard output

In the probability theory the following paradox called Benford's law is known: "In many lists of random numbers taken from real sources, numbers starting with digit 1 occur much more often than numbers starting with any other digit" (that's the simplest form of the law).

Having read about it on Codeforces, the Hedgehog got intrigued by the statement and wishes to thoroughly explore it. He finds the following similar problem interesting in particular: there are N random variables, the i-th of which can take any integer value from some segment [Li;Ri] (all numbers from this segment are equiprobable). It means that the value of the i-th quantity can be equal to any integer number from a given interval [Li;Ri] with probability 1 / (Ri - Li + 1).

The Hedgehog wants to know the probability of the event that the first digits of at least K% of those values will be equal to one. In other words, let us consider some set of fixed values of these random variables and leave only the first digit (the MSD — most significant digit) of each value. Then let's count how many times the digit 1 is encountered and if it is encountered in at least K per cent of those N values, than such set of values will be called a good one. You have to find the probability that a set of values of the given random variables will be a good one.

Input

The first line contains number N which is the number of random variables (1 ≤ N ≤ 1000). Then follow N lines containing pairs of numbers Li, Ri, each of whom is a description of a random variable. It is guaranteed that 1 ≤ Li ≤ Ri ≤ 1018.

The last line contains an integer K (0 ≤ K ≤ 100).

All the numbers in the input file are integers.

Please, do not use %lld specificator to read or write 64-bit integers in C++. It is preffered to use cin (also you may use %I64d).

Output

Print the required probability. Print the fractional number with such a precision that the relative or absolute error of the result won't exceed 10 - 9.

Examples
input
Copy
1
1 2
50
output
Copy
0.500000000000000
input
Copy
2
1 2
9 11
50
output
Copy
0.833333333333333



 

Code Examples

#1 Code Example with C++ Programming

Code - C++ Programming

#include <bits/stdc++.h>

using namespace std;

int const N = 1e3 + 1;
int n, k;
long long l[N], r[N], ones[N], gl[19], gr[19];
double dp[N][N];

int first_digit(long long x) {
  int ret;
  while(x != 0)
    ret = x % 10, x /= 10;
  return ret;
}

long long bf(int i) {
  long long ret = 0;
  for(long long j = l[i]; j <= r[i]; ++j)
    if(first_digit(j) == 1)
      ++ret;
  return ret;
}

long long calc(int i) {
  long long ret = 0;
  for(int j = 0; j < 19; ++j) {
    if(gl[j] >= l[i] && gr[j] <= r[i])
      ret += gr[j] - gl[j] + 1;
    else if(gl[j] >= l[i] && gl[j] <= r[i])
      ret += r[i] - gl[j] + 1;
    else if(gr[j] >= l[i] && gr[j] <= r[i])
      ret += gr[j] - l[i] + 1;
    else if(l[i] >= gl[j] && r[i] <= gr[j])
      ret += r[i] - l[i] + 1;
  }
  return ret;
}

double rec(int idx, int cnt) {
  if(idx == n)
    return (1.0 * cnt / n * 100.0) >= k;

  double &ret = dp[idx][cnt];
  if(ret == ret)
    return ret;
  ret = 0;

  return ret = 1.0 * ones[idx] / (r[idx] - l[idx] + 1) * rec(idx + 1, cnt + 1) + 1.0 * ((r[idx] - l[idx] + 1) - ones[idx]) / (r[idx] - l[idx] + 1) * rec(idx + 1, cnt);
}

int main() {
  gl[0] = gr[0] = 1;
  for(int i = 1; i < 19; ++i) {
    gl[i] = gl[i - 1] * 10;
    gr[i] = gl[i] * 2 - 1;
  }

  scanf("%d", &n);
  for(int i = 0; i < n; ++i) {
    scanf("%lld %lld", l + i, r + i);
    ones[i] = calc(i);
    // cout << ones[i] << ' ' << bf(i) << endl;
  }
  scanf("%d", &k);

  memset(dp, -1, sizeof dp);
  printf("%0.10lf\n", rec(0, 0));

  return 0;
}
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Input

x
–
+
cmd
1
1 2
50

Output

x
–
+
cmd
0.500000000000000
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Demonstration

Codeforcess Solution First Digit Law ,C. First Digit Law C,C++, Java, Js and Python ,C. First Digit Law,Codeforcess Solution

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