This exercise is about the well-known Monty Hall problem. In the problem, you are a con testant on a game show. The host, Monty Hall, shows you three doors. Behind one of those doors is a prize, and behind the other two doors are goats. You pick a door. Monty Hall, who knows behind which door the prize lies, then opens up one of the doors that doesn’t contain the prize. There are now two doors left, and Monty gives you the opportunity to change your choice. Should you keep the same door, change doors, or does it not matter ?

(a) Write a program that simulates playing this game 10000 times and calculates what per�centage of the time you would win if you switch and what percentage of the time you would win by not switching.

        import random

        def monty_hall_simulation(num_simulations):
            stay_wins = 0
            switch_wins = 0
        
            for _ in range(num_simulations):
                # Randomly place the prize behind one of the three doors
                doors = ["goat", "goat", "prize"]
                random.shuffle(doors)
        
                # Contestant makes an initial choice
                initial_choice = random.randint(0, 2)
        
                # Monty opens one of the other doors with a goat
                monty_choices = [i for i in range(3) if i != initial_choice and doors[i] == "goat"]
                monty_open = random.choice(monty_choices)
        
                # Contestant's options after Monty's revelation
                remaining_choices = [i for i in range(3) if i != initial_choice and i != monty_open]
        
                # Determine outcomes for staying and switching
                if doors[initial_choice] == "prize":
                    stay_wins += 1
                else:
                    switch_wins += 1
        
            stay_percentage = (stay_wins / num_simulations) * 100
            switch_percentage = (switch_wins / num_simulations) * 100
        
            print(f"Simulations: {num_simulations}")
            print(f"Percentage of wins by staying: {stay_percentage:.2f}%")
            print(f"Percentage of wins by switching: {switch_percentage:.2f}%")
        
        if __name__ == "__main__":
            num_simulations = 10000
            monty_hall_simulation(num_simulations)
        
    

(b) Try the above but with four doors instead of three. There is still only one prize, and Monty still opens up one door and then gives you the opportunity to switch.

        import random

        def monty_hall_simulation_four_doors(num_simulations):
            stay_wins = 0
            switch_wins = 0
        
            for _ in range(num_simulations):
                # Randomly place the prize behind one of the four doors
                doors = ["goat", "goat", "goat", "prize"]
                random.shuffle(doors)
        
                # Contestant makes an initial choice
                initial_choice = random.randint(0, 3)
        
                # Monty opens one of the other doors with a goat
                monty_choices = [i for i in range(4) if i != initial_choice and doors[i] == "goat"]
                monty_open = random.choice(monty_choices)
        
                # Contestant's options after Monty's revelation
                remaining_choices = [i for i in range(4) if i != initial_choice and i != monty_open]
        
                # Determine outcomes for staying and switching
                if doors[initial_choice] == "prize":
                    stay_wins += 1
                else:
                    switch_wins += 1
        
            stay_percentage = (stay_wins / num_simulations) * 100
            switch_percentage = (switch_wins / num_simulations) * 100
        
            print(f"Simulations: {num_simulations}")
            print(f"Percentage of wins by staying: {stay_percentage:.2f}%")
            print(f"Percentage of wins by switching: {switch_percentage:.2f}%")
        
        if __name__ == "__main__":
            num_simulations = 10000
            monty_hall_simulation_four_doors(num_simulations)
        
    

These programs simulate the Monty Hall problem for 10,000 iterations and calculate the percentage of wins when staying and when switching doors, for both three and four doors scenarios.