[thehappypm]

I don’t think this helps me understand it.

In the 1,000 doors problem, my odds of being right initially were something like 1/1000 and then it changes to something like 998/1000 or 999/1000 for switching, I can’t intuitively grasp exactly what the odds become of winning if I switch, I just know it’s high. Bringing it down to 3 doors doesn’t help me much — it’s still something like 1/2 or 1/3.

[cestith]

Try thinking of it this way. It may or may not be any more useful. I've seen different people come to understand the problem from different examples.

    1. Observe that 3/3 = 1. Pedantic, yes, but good for frame of mind here.
    2. Pick one of three doors. (1/3 odds)
    3. Gain information that one of the three doors is a loser.
    4. Note your odds on choosing the original door correctly are still 1/3.
    5. Note that if you change doors, there are still 2/3 doors there to choose.
    5. Note you're not going to switch to the known loser door, so if you change doors you know 100% which of the other 2/3 of doors to choose.
  

The intuition usually is that you're down to two doors after the loser door is opened, but that's not the case. There are still three doors. The host has just told you that if you trade doors, you know which door to trade for. So trade for it.

Note there's a newer version of "Let's Make a Deal", hosted by Wayne Brady, but there is no option to switch after a losing door has been shown in that version.

[thehappypm]

Step 4, kind of a mystery. Why are my odds still the same even though the situation is different?

[inetknght]

You choosing a door in the first place gave you 1-in-3 chance. But your 1-in-3 choice is deducted from the 3-of-3 choices that Monty could have had, so Monty only had a 2-in-3 choice: there's a 2/3 chance that the car is in the pool of doors from which Monty could select. Monty has a 100% chance of choosing a door without a car. Therefore, you inherit the 2-in-3 chances if you change your selection.

The first door will have the car 1/3 of the time. The second door's chances had been expanded to the remaining 2/3 percent thanks to Monty always choosing the last 1/3 door which is guaranteed to not have a car.

[Swenrekcah]

Because it isn't different really. It is always a goat door that is opened, so you don't gain any information about your door by the opening of the goat door.

I'm thinking of a number between 1 and 10, guess it. If I now tell you a number I promise is not the one I was thinking and not your number, you have no more information about if you were correct.

[SamBam]

Because it really centers on the initial premise: Monty will always open a goat door after your choice, no matter what.

So, you make a totally random choice. That choice must be 1/3 right, right? Now the thing that you already knew would definitely happen happens: Monty opens a goat door. How can your odds suddenly jump to 1/2?

Are you saying every single time you play the game, you always have a 1/2 chance of getting it right first time?

[cestith]

What you knew about the door you initially picked hasn't changed at all. What you now know about the other doors has. By giving you information about 1/2 of the other 2/3 doors, Monty has given you an extra 1/3 chance if you pick among those.

[bartc]

Assume there are seven billion people on the planet. One of them knows the location of a specific hidden treasure. I know who it is and I ask you to guess who it is, but you have no possible way of knowing or even getting a hint about it.

You pick some random person. I then bring in another stranger and tell you that the person who knows where the hidden treasure is is either the random person you chose or the one I brought in.

At this point, there are only two possibilities:

1. You happened to randomly choose the right person on Earth and in my surprise, I had to pick some other random stranger to pretend they knew the secret.

2. You chose a total rando who has no idea what's going on and the person I brought in is in fact the one who knows where the treasure is

[ragnese]

You don't need to know the exact odds to understand that it's higher. I think that's the main takeaway of making it a 1,000 door problem. It makes it intuitive that the correct solution is to switch. The exact probability doesn't matter.

[delecti]

IMO what helps is to imagine slightly changing the order.

First step is still that you pick a door. There's a 1/3 chance it has the car. Now you can either keep that single door (with a 1/3 chance of a car), or switch and get both of the other two doors (each with a 1/3 chance of the car, for a total of 2/3 chance). After you pick, I'll reveal all the goats.

[klmadfejno]

probability of winning if you switch is 1 - (1 / n) aka (n-1)/n.

probability of winning if you don't is 1 / n.

By switching, you are simply betting that your original guess of 1/n was wrong.

[Closi]

Try watching this video below, which explains the 1/100 analogy with a real world example:

https://youtu.be/GPoPSNxV1D4?t=365

I've timestamped the relevant bit - but you should watch the full thing from the start, it's very entertaining :)