Monty Hall Problem Intuition

Question

I was thinking about the Monty Hall problem and I thought of a possible intuitive explanation:

• You choose a door.
• Monty gives you the option of sticking with your original choice or instead choosing both of the other two doors.
• If you decided to switch (which now becomes an obvious choice), Monty first opens the door with the goat behind it (say, to add to the excitement), and then opens the other door.

My question then is, is this reasoning flawed? Is this even the same problem as before? Because now, choosing to switch from one door to two doors becomes quite obvious, and so does the $2/3^{rd}$ chance of winning the car on switching.

Answer 1

Yes this is the same problem. Since in the original solution Monty opens one door, and thus that one is bad, hence choosing that one too does not make a difference.

My favourite way of convincing people is to, instead of considering 3 doors, consider 1000 doors. If, after you have choosen 1 door, Monty opens all other doors except one, then obviously you will switch.

Answer 2

I believe the best way to intuitively understand the Monty Hall problem is by playing the game with a $100$ doors, $99$ goats and one supercar.

I can choose a door, doing so will give me a probability of $1\%$ of choosing the car. The host then opens $98$ doors, showing $98$ goats. At this point I know that the door I chose either contains the supercar (with a probability of $1\%$) or more likely a goat ($99\%$).

Now I'm given the oppurtinity to switch doors, it's clear that doing so will increase my chance of getting the supercar to $99\%$.

Answer 3

I cannot find a flaw in your reasoning.

My own reasoning (if you are interested).

Someone who sticks to his original choice will win if his original choice was correct.

Probability on that: $\frac13$.

Someone who switches will win if his original choice was wrong.

Probability on that: $\frac23$.

Answer 4

Here is the way I convinced myself of the truth that: "the odds of winning the car by switching the original choice is 2/3," given that the host knows the locations of the car and the two goats in advance (stated fact in the problem.)

I changed my thinking to consider that the host is not choosing a DOOR, but rather he is choosing a PRIZE...i.e. he will always pick a goat-door in every case, regardless of its number (assumption that must be inferred because we must assume he won't give the car away by opening its door.) I think much confusion is generated by the wording in the problem (and much of the debates) that the contestant chooses "Door 1" and the host chooses "Door 3."

Then when we look at the three possible arrangements of prizes and doors, and all possible door-choice and prize-choice sequences (see matrices elsewhere), the 2/3 probability becomes clearer.

Answer 5

In my opinion this intution is not quite right. you say that the incident "the prize is behind door 1 OR door 2" is equivalent to switching to doors 1 OR 2 when the instructor opens, lets say, door number 1 and reveals a goat.

but when the instructor reveals a goat behind door number 1, the probability of the incident "the prize is behind door 1 OR door 2" is not the same as before, it shouldn't stay 66% because there is new information we added to the game.

so pure intution - this is not a convincing one in my opinion.

more convincing(imo) is this one: if the prize is behind the door i chose - I DONT WANT TO SWITCH, but the probability that the prize is behind the door i chose is exactly 1/3. and the probability that the prize is behind any other door is exactly 2/3. in this case, I WANT TO SWITCH. this means that in 66% of the cases you would need to switch doors in order th win, and in 33% of the cases you would need to stay with your choice in order to win.

Answer 6

The way I understand the solution is:

It's easier to select the wrong prize on the first pick. Here(2/3). Now, after more information is pumped in by the game show host, it becomes certain that a switch would ensure the prize(if you selected a goat, and the host shows the next goat, the you are certain to win the prize on switch). So you get a 2/3 * 1 chance of winning the prize if you switch.

The probability of a switch relies on the fact that it was easier to make the wrong choice in the first go.