from random import randint import matplotlib.pyplot as plt import matplotlib.ticker as ticker
This function will generate random rounds for our game. Each round consists of 3 doors. Only one of the doors is correct, other two are wrong.
def generate_game(n: int): game =  for _ in range(n): doors = [False] * 3 winner = randint(0, 2) # Choose winning door by random doors[winner] = True game.append(doors) return game
This is a helper function that takes a list of 3 doors, looks at the second and third door, and then opens the one with goat (i.e. wrong door). This simulates a host with knowledge of what is behind the doors.
def reveal_goat(doors): # Get from doors 2 and 3 the one which contains goat. for i in range(1, 3): if doors[i] == False: return i
Simulate random choice
Simulate situation where the player randomly chooses whether to keep his initial choice or switch his choice.
def simulate_random_choice(game: list): wins = 0 attempts = 0 history =  for doors in game: attempts += 1 # Host reveals a door with goat. goat = reveal_goat(doors) # Player randomly chooses whether to keep initial choice or switch. new_choice = randint(0, 1) final_choice = 0 if new_choice == 0 else 2 if goat == 1 else 1 if (doors[final_choice] == True): wins += 1 history.append(wins / attempts) return wins, history
Simulate initial choice
Simulate situation where the player only keeps his initial choice and never switches.
def simulate_keep_choice(game: list): wins = 0 attempts = 0 history =  for doors in game: attempts += 1 # User does not switch game. if (doors == True): wins += 1 history.append(wins / attempts) return wins, history
Simulate switch choice
Simulate situation where the player switches his choice everytime.
def simulate_switch_choice(game: list): wins = 0 attempts = 0 history =  for doors in game: attempts += 1 # Host reveals a door with goat. goat = reveal_goat(doors) # Player switches his doors (here he chooses the non-opened doors). new_choice = 1 if goat == 2 else 2 if (doors[new_choice] == True): wins += 1 history.append(wins / attempts) return wins, history
Now the computing begins. This generates $n$ random games for simulation.
game = generate_game(1000)
Run the three simulations defined above for the generated game.
wins_random, history_random = simulate_random_choice(game) wins_keep, history_keep = simulate_keep_choice(game) wins_switch, history_switch = simulate_switch_choice(game)
And finally, create some fancy graph so we can actually see the result.
plt.figure(figsize=(12,8)) plt.plot(history_random, 'r', label="Random switch") plt.plot(history_keep, 'g', label="Keep initial") plt.plot(history_switch, 'b', label="Only switch") plt.legend(loc='upper right') plt.ylim(0, 1.0) plt.xlim(0, 1000) plt.ylabel("Chance", fontsize=16) plt.gca().yaxis.set_major_formatter(ticker.PercentFormatter(xmax=1.0)) plt.xlabel("Iterations", fontsize=16) plt.grid(True) plt.show()