Marilyn vos Savant and the Monty Hall Problem (2015)

A nice goat fetches up to $1000 to according to a quick google.

The idea is supposed to be that Monty always reveals a goat. He does not pick a door at random to reveal before offering the choice to switch. If you pick door A, and the car is in door B, he will reveal door C and ask if you want to switch. If you pick A, the car is in C, then he will open door B and ask if you want to switch.

Expanding it to 100 doors with Monty opening 98 of them and then offering you the opportunity to switch should make it very intuitive. Let's say you pick door 7, and Monty then opens every door except your door and say, door 48, all revealing goats, then it should make you think "Wow...very odd that he didn't reveal door 48...there must be a reason he skipped that specific door".

Make it 1,000,000 doors. You choose door 1. Monty opens every door except number 423,901, and all the doors he opened had goats. You're tired and hungry because all you've done for 4 days is watch Monty open doors to goats, but you should certainly be thinking it's odd that he skipped that one specific door. You should probably switch, though maybe take a nap before claiming your car.

So I see the second order problem a bit differently: it's not "was the statement unambiguous", it's about what you do with the (possibility of) ambiguity: do you assume you can't be wrong and therefore others must be, or do you try to figure out whether you were solving different problems?

edit: i ran a monty carlo simulation¹ and i was doing the math wrong. it really does matter if monty knows or not. here's the simulation where he knows:

    In [15]: non_censored_trials = got_car_trials = 0

    In [16]: for trial in range(100_000):
        ...:     car_door = random.randrange(3)  # the other two doors have goats
        ...:     your_door = random.randrange(3)
        ...:     monty_door = random.choice(list({0, 1, 2} - {your_door, car_door}))
        ...:     if monty_door == car_door:
        ...:         print("Monty showed the car, never mind")
        ...:         continue
        ...:     non_censored_trials += 1
        ...:     your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door}))  # you change your choice
        ...:     if your_new_choice == car_door:
        ...:         print(f"You got the car because you changed from {your_door} to {your_new_choice}")
        ...:         got_car_trials += 1
        ...:     else:
        ...:         print(f"Too bad you changed; you should have stuck with {your_door}")

    (...output omitted...)

    In [17]: got_car_trials / non_censored_trials
    Out[17]: 0.66921

    In [21]: non_censored_trials = got_car_trials = 0

    In [22]: for trial in range(100_000):
        ...:     car_door = random.randrange(3)  # the other two doors have goats
        ...:     your_door = random.randrange(3)
        ...:     monty_door = random.choice(list({0, 1, 2} - {your_door}))
        ...:     if monty_door == car_door:
        ...:         print("Monty showed the car, never mind")
        ...:         continue
        ...:     non_censored_trials += 1
        ...:     your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door}))  # you change your choice
        ...:     if your_new_choice == car_door:
        ...:         print(f"You got the car because you changed from {your_door} to {your_new_choice}")
        ...:         got_car_trials += 1
        ...:     else:
        ...:         print(f"Too bad you changed; you should have stuck with {your_door}")

    (...output omitted...)

    In [23]: got_car_trials / non_censored_trials
    Out[23]: 0.49987257709086

______

¹ thank you, manoj