Monty Hall Problem Re-framed

Monty Hall Problem Re-framed


Embarrassingly enough, it took me 2 years since first hearing the Monty Hall problem to understand the answer. I would oscillate between a feeling of being close to an Ah! moment, and utter loss at what is going on.

Wikipedia has the essence of the problem phrased as such,

Suppose you're on a game show, and you're given the choice of three  doors: Behind one door is a car; behind the others, goats. You pick a  door, say No. 1, and the host, who knows what's behind the doors, opens  another door, say No. 3, which has a goat. He then says to you, "Do you  want to pick door No. 2?" Is it to your advantage to switch your choice?
vos Savant, Marilyn (9 September 1990a). "Ask Marilyn". Parade Magazine: 16.

The intuitive answer for many, including myself initially, is to say it makes no difference whether you switch or you don't. Obviously, this question become popular because the intuitive answer is the incorrect one.

After struggling with it for a while, I finally understood the answer by switching around and rephrasing the question so that instead of asking the question "Is it to your advantage to switch?", I ask "How likely is it that you picked the right door?".

Given 3 sets of doors, there are 9 configurations possible.

Your Selection Door with Car
1 1
1 2
1 3
2 1
2 2
2 3
3 1
3 2
3 3

With the above data, the question of "Is it to your advantage to switch?" is the same as answering "How likely is it that you picked the right door?". This can easily be answered by showing what action (switching or not switching) will lead to a win in each configuration.

Your Selection Door with Car Winning Case
1 1 Not Switch
1 2 Switch
1 3 Switch
2 1 Switch
2 2 Not Switch
2 3 Switch
3 1 Switch
3 2 Switch
3 3 Not Switch

You can see that only 1/3 of the cases end well if you don't switch. The rest work out only if you switch.

Sometime intuition is horrible at guessing the outcomes of certain type of situations, and this, for chumps like myself, is a great demonstration.