The Monty Hall Problem

monty.rb

# Simulation of the Monty Hall problem
#
# In a given game, there are three doors with a prize behind one.
# The player picks one of the doors.
# Monty removes one of the non-selected doors that isn't the prize door.
# There are now two doors left in play, and one contains a prize.
# Monty asks the player if they'd like to switch doors.
#
# What should the player do? What is the probability the player will win if
# they switch doors vs staying the same?
#
# The program randomly simulates the results of many rounds of the game to see
# what the probabilities are like.

#
# Game
#
# This class is responsible for selecting the doors and detecting if the game
# was won by the player.
#
class Game
  def initialize
    won?
  end

  # Compare the prize door to either the new door or original door depending on
  # whether the player switched or not.
  def won?
    if switched?
      new_door == prize_door
    else
      original_door == prize_door
    end
  end

  def switched?
    defined?(@sw) ? @sw : @sw = [true, false].sample
  end

  def original_door
    @od ||= random_door
  end

  def prize_door
    @pd ||= random_door
  end

  def new_door
    @nd ||= begin
              # The new door cannot be the original door
              doors = [0, 1, 2] - [original_door]

              # Remove one of the doors that isn't the original or prize door.
              doors -= ([0, 1, 2] - [original_door, prize_door])

              # Pick one of the remaining doors
              doors.sample
            end
  end

  def random_door
    rand(3)
  end

  def inspect
    "[ #{door_ind(0)} ][ #{door_ind(1)} ][ #{door_ind(2)} ] #{'Won!' if won?}"
  end

  def to_s
    inspect
  end

  private

  def door_ind(index)
    i = ''
    i << 'O' if index == original_door
    i << 'N' if siwtched? && index == new_door
    i << 'P' if index == prize_door
    i
  end
end

def print_report(set)
  won = set.select(&:won?).count
  puts "total: #{set.count}"
  puts "won: #{won} (#{(won.to_f / set.count * 100).round(1)}%)"
  puts ''
end

games = Array.new(10_000).map { Game.new }

switched = games.select(&:switched?)
stayed = games - switched

puts 'Switched'
puts print_report(switched)

puts 'Stayed'
puts print_report(stayed)

# The expected output reads something like the following
#
#     Switched
#     total: 5065
#     won: 3372 (66.6%)
#
#     Stayed
#     total: 4935
#     won: 1656 (33.6%)
#
# This may be a surprising result, that switching your door more likely leads
# to a win, however this is normal probabilities.
#
# An intuitive explanation is that switching wins if you picked a bad door.
# It's more likely that you picked a bad door than the right one, so it's more
# likely that switching will win.