dice_probabilities.rb

def throw_die(count, sides)
  # Throw die randomly and add them up
  count.times.sum{ rand(1..sides) }
end

def simulate(m, n, sides)
  k = 100000

  wins = k.times.count{ 
    throw_die(m, sides) > throw_die(n, sides)
  }

  wins.fdiv(k).round(2)
end

def calculate(m, n, sides)
  m_probs = probabilities(m, sides)
  n_probs = probabilities(n, sides)

  # Calculate the probability of the sum of 
  # m-die being greater than the sum of n-die
  m_probs.map.with_index(-1){ |prob, i|
    prob * n_probs[..i.clamp(0..)].sum 
  }.sum.round(2)
end

def probabilities(count, sides)
  if count == 1
    return [1.fdiv(sides)] * sides
  end

  previous_probs = probabilities(count - 1, sides)

  # Shift the probabilities to account for the new possible results
  shifted_probs = sides.times.map{ |number| 
    prefix = Array.new(number + 1, 0)
    sufix = Array.new(sides - number - 1, 0)
    prefix + previous_probs + sufix
  }

  # Update the probability of every result
  shifted_probs.transpose.map{ |number_probs| number_probs.sum / sides }
end

puts simulate(4, 3, 6)
puts calculate(4, 3, 6)