Markov chain simulation

This problem was asked by Google.

You are given a starting state start, a list of transition probabilities for a Markov chain, and a number of steps num_steps. Run the Markov chain starting from start for num_steps and compute the number of times we visited each state.

For example, given the starting state a, number of steps 5000, and the following transition probabilities:

[
  ('a', 'a', 0.9),
  ('a', 'b', 0.075),
  ('a', 'c', 0.025),
  ('b', 'a', 0.15),
  ('b', 'b', 0.8),
  ('b', 'c', 0.05),
  ('c', 'a', 0.25),
  ('c', 'b', 0.25),
  ('c', 'c', 0.5)
]

One instance of running this Markov chain might produce { 'a': 3012, 'b': 1656, 'c': 332 }.

My Solution(C++):

#include <iostream>
#include <map>
#include <vector>
#include <cstdlib>
#include <ctime>
#define mp std::make_pair

void markov_chain(std::map<char, std::vector<std::pair<char, double>>> H, int num_steps, char start){
  std::srand(std::time(nullptr));
  std::map<char, int> freq;
  for (int i=0; i<num_steps; i++){
    double p = double(rand()%1000)/double(1000);
    // std::cout<<p<<'\n';
    double cum_prob = 0;
    for (auto entry: H[start]){ // the  pairs
      cum_prob+=entry.second;
      if (p<cum_prob){
        freq[entry.first]++;
        start = entry.first;
        break;
      }
    }
  }
  for (auto x: freq) std::cout<<x.first<<':'<<x.second<<'\n';
}

void test(){
  std::map<char, std::vector<std::pair<char, double>>> H;
  H['a'] = {mp('a', 0.9), mp('b', 0.075), mp('c', 0.025)};
  H['b'] = {mp('a', 0.15), mp('b', 0.8), mp('c', 0.05)};
  H['c'] = {mp('a', 0.25), mp('b', 0.25), mp('c', 0.5)};
  markov_chain(H, 5000, 'a');
}

int main(){
  test();
  return 0;
}