My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <vector>
       
 
       
typedef unsigned long long Possibilities; // replace unsigned long long by double to compute up to 30 rounds
       
 
       
struct State
       
{
       
  // picked red and blue disc
       
  unsigned int red;
       
  unsigned int blue;
       
  // number of different choices leading to this state
       
  Possibilities possibilities;
       
};
       
 
       
int main()
       
{
       
  unsigned int tests = 1;
       
  std::cin >> tests;
       
 
       
  while (tests--)
       
  {
       
    unsigned int maxRounds = 15;
       
    std::cin >> maxRounds;
       
 
       
    // number of red and blue discs in the bag
       
    unsigned int availableRed   = 1;
       
    unsigned int availableBlue  = 1;
       
    Possibilities possibilities = 1;
       
 
       
    // initially no disc taken yet
       
    std::vector<State> open = { { 0, 0, 1 } };
       
 
       
    // evaluateall 15 rounds
       
    for (unsigned int round = 1; round <= maxRounds; round++)
       
    {
       
      // total number of possible choices
       
      possibilities *= availableRed + availableBlue;
       
 
       
      // set number of blue and red discs per state
       
      std::vector<State> next;
       
      for (unsigned int blue = 0; blue <= round; blue++)
       
      {
       
        State state;
       
        state.red  = round - blue;
       
        state.blue = blue;
       
        state.possibilities = 0;
       
        next.push_back(state);
       
      }
       
 
       
      // update possibilities of each state
       
      for (auto current : open)
       
      {
       
        // picking red
       
        next[current.blue    ].possibilities += current.possibilities * availableRed;
       
        // picking blue
       
        next[current.blue + 1].possibilities += current.possibilities * availableBlue;
       
      }
       
 
       
      // prepare next iteration
       
      open = next;
       
      availableRed++;
       
    }
       
 
       
    // compute chance that blue > red
       
    Possibilities moreBlue = 0;
       
    for (auto current : open)
       
      if (current.blue > current.red)
       
        moreBlue += current.possibilities;
       
 
       
    // chance of winning is moreBlue / possibilities, therefore maximum prize is the inverse
       
    unsigned int prize = possibilities / moreBlue; // integer division !
       
    std::cout << prize << std::endl;
       
  }
       
 
       
  return 0;
       
}

This solution contains 13 empty lines, 13 comments and 2 preprocessor commands.