This was the one that worked when explaining it to my friends.

It gives a mental image of the host opening 998 boxes, leaving only your selected box and one other. From here it’s easier to see that there must be something special about that one box the host left un-opened!

(Though even then there were people who clung to the “2 boxes means 1-in-2 chance” fallacy, failing to see that the host has revealed information.)

Edit: an other version was to change the hosts proposal: what if he let you choose one box, and then said he would let you switch to having whatever was in the other 999 boxes? Of course you would switch! The crux is understanding that this offer is actually the same as in the first proposal, since the host is not opening the boxes at random.

To me, THAT is the most powerful intuitive description.

This doesn't do anything for me.

(I understand the Monty Hall problem, I just don't see how changing the number of doors makes a difference to anyone's intuition.)

It's because it makes the initial choice so increasingly unlikely (increasing with the number of doors) to be correct that when the doors are taken away and you're left with only two, one of which must be right, it means that the other door is incredibly likely to be the right one.

> It's because it makes the initial choice so increasingly unlikely

But you still need to conivnce people that the one remaining unopened door is more likely than the door you originally selected. They were both unlikely to begin with, ramping up the number of doors doesn't explaing why one of them should be preferred.

There are 1000 doors. You choose 1. Anyone knows it's incredibly unlikely that the correct one is chosen first time.

Now 998 doors are removed. There is 1 door from the others and the door you choose. Given that your choice is almost certainly wrong, and that your opponent couldn't remove the correct door from amongst the 998, that means the other door is the correct one.

Is that convincing enough?

Because a 99/100 chance is much better than 2/3 to drive the point home...

Imagine there are 999 boxes with nothing in them and one box with the keys. After picking a box, the hosts opens 998 empty boxes. Would you still stick with your initial choice?

I would change my choice because I understand the problem. But I would also change my choice in the scenario with 3 boxes. I'm not arguing with the conclusion, what I don't understand is the people who have their mind changed by the argument.

Extending it to 1000 boxes/doors still doesn't explain why the remaining unopened box is different from the box your picked originally.

> the remaining unopened box is different from the box your picked originally.

Because you now know that every other box is empty. So by process of elimination you know that your box and the remaining one are different.

I've often found it easier to understand things intuitively by putting an idea to the slippery slope test. If such & such were true, imagine changing some parameters to an extreme, how absurd does it become? For monotonic functions it's useful

And the key thing here is that all boxes except two gets removed, not only one.

That was a new spin to the explanation that I didn't think of before.