Below is a table showing the probabilities of different outcomes in Risk.  The probabilities were calculated using a Matlab function which is listed below the table.  For each die roll combination, one million trials were excecuted to ensure smooth statistics.
 

function prob = RiskRoll(trials, attack, defend)

% Computes the probabilities for rolls in Risk.
% trials represents the number of times the function
% will excecute the rolls and calculate outcome.  Thus
% the more trials, the better the statistic.  attack
% is the number of attack dice being rolled, and defend
% is the number of defense die being rolled.

% prob is a 5 row vector that contains the probabilities
% of the follwing occurring...
% [Lose one each
%  Attacker loses one
%  Defender loses one
%  Attacker loses two
%  Defender loses two]

% First initialize variables...
OneEach = 0;
ALoseOne = 0;
DLoseOne = 0;
ALoseTwo = 0;
DLoseTwo = 0;

% loop through the trials, summing the different results...

for i = 1:trials

% to keep a count of pieces lost each trial...

 ALoses = 0;
 DLoses = 0;

% computes uniformly distributed random numbers for the
% attack and defense rolls...

 ADie = floor(6*rand(attack,1)+1);
 DDie = floor(6*rand(defend,1)+1);

% sorts the vectors from greatest to least...

 ADie = -sort(-ADie);
 DDie = -sort(-DDie);

% the following two if statements find the maximum value(s)
% of the rolls...

 if (attack >= 2)
  AMax1 = ADie(1);
  AMax2 = ADie(2);
 else
  AMax1 = ADie(1);
  AMax2 = 0;
 end

 if (defend >= 2)
  DMax1 = DDie(1);
  DMax2 = DDie(2);
 else
  DMax1 = DDie(1);
  DMax2 = 0;
 end

% computes the number lost for each side during the current
% trial...

 if ((attack >= 2) & (defend >= 2))
  if (DMax1 >= AMax1)
   ALoses = 1 + ALoses;
  else
   DLoses = 1 + DLoses;
  end
  if (DMax2 >= AMax2)
   ALoses = 1 + ALoses;
  else
   DLoses = 1 + DLoses;
  end
 else
  if (DMax1 >= AMax1)
   ALoses = 1 + ALoses;
  else
   DLoses = 1 + DLoses;
  end
 end

% the following if statement finds the appropriate variable to
% increment based on the troops lost...

 if (ALoses == 1) & (DLoses == 0)
  ALoseOne = 1 + ALoseOne;
 elseif (ALoses == 1) & (DLoses == 1)
  OneEach = 1 + OneEach;
 elseif (ALoses == 0) & (DLoses == 1)
  DLoseOne = 1 + DLoseOne;
 elseif (ALoses == 2)
  ALoseTwo = 1 + ALoseTwo;
 else
  DLoseTwo = 1 + DLoseTwo;
 end

end

% takes the average...

OneEach = OneEach/trials;
ALoseOne = ALoseOne/trials;
DLoseOne = DLoseOne/trials;
ALoseTwo = ALoseTwo/trials;
DLoseTwo = DLoseTwo/trials;

prob = [OneEach;ALoseOne;DLoseOne;ALoseTwo;DLoseTwo;];