I see this posted a lot but it doesn't click with me. It only feels obvious once I know how the probabilities work - meaning I have to get the base case first.
Picture, in your mind or maybe draw on paper, 20 brightly lit doors in a row where 19 are labeled L and one is labeled W. Out of those 20 doors, one door gets randomly selected. It's almost certainly going to be an L. After choosing it, the lights go out above 18 of the other L doors so only the W door, and the L door you probably chose, remained lit.
If you stayed with your existing selection— one of two doors that remains lit— you've still almost certainly got an L door. If you switch to the one other door, with all of the other L doors eliminated, the only way it's not the W door is if you chose the W to begin with.
1. It requires that I already get the point about the probabilities.
2. You don't get a benefit in Monty hall if the host picks randomly - the description requires that they deliberately leave the winning door available. Any situation that gives you the same intuition but you just picture as happening without intent has given you the wrong feeling.
I don't find that increasing the number of doors makes it feel any different.
Yes. The original phrasing is ambiguous though and the one I replied to doesn't say this. Without this very specific point, the other intuitive answer about switching is actually correct.
> Picking randomly wouldn’t make sense. You’d pick a goat, then the host could randomly reveal the car and the game would end anticlimactically
If you want to try and take the riddle that way, it also wouldn't make sense to always offer the switch. They offered the switch this time, but is that a bluff? Would make for a lot better tv show.
But just like prison guards don't set up elaborate schemes for 100 prisoners to be let go, we constrain ourselves to the invented universe of the riddle.
If you read a framing of lights with l and w and no setup explaining how the lights are logically changed the answer is not clear. If it is - your intuition is wrong.
The original phrasing of the problem is not ambiguous at all:
> 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?
You can change the rules however you like but then it’s no longer the Monty Hall problem, and that’s what we’re discussing: a brain teaser with well defined rules and probabilistic interpretation.
That's not the original phrasing because that's from 1990, have a look at the wiki page you linked. The original problem does not specify that Monty will never open a box with the keys in. The game would still work if this was the case as Monty offers to buy the box before doing so. In the original problem, Monty says the chance is 1/2, which would actually be true if they had opened them randomly. (edit - a game where the host offers to buy the box you have before another one is opened is essentially the end part of deal or no deal).
Even the 1990 version however is ambiguous when it comes to puzzles. It just says they open a door which has a goat.
The solution originally posted explicitly assumes it to be the case, which is fine.
Exactly— the entire point is that the host always eliminates every losing answer that you didn't choose unless you chose the right answer initially. If you keep your pick, you have a 1 in 3 chance of winning, and if you switch, you have a 1 in 3 chance of losing.
I didn't simply increase the number of doors— I created a visual of a more extreme version of the problem without making it so extreme that it's abstract. That sort of thing helps a lot of visual thinkers and I'm not sure why you're being so pissy about it not helping you. Frankly, I'm sorry I tried.
I'm not being pissy about it, I'm explaining why this approach doesn't click for me. I'm not sure why, after being told that this type of explanation doesn't work for me and being told exactly why, you are surprised when I say that again. I'll try and help explain why in more detail.
The point is that for me the key bit of information is yet again missing from your explanation.
> Picture, in your mind or maybe draw on paper, 20 brightly lit doors in a row where 19 are labeled L and one is labeled W. Out of those 20 doors, one door gets randomly selected. It's almost certainly going to be an L. After choosing it, the lights go out above 18 of the other L doors so only the W door, and the L door you probably chose, remained lit.
It's extremely important the reasoning behind why the 18 other doors are picked is highlighted.
The game show deal or no deal has basically this, but with 20 or 25 or something boxes. At the end they get to switch - and it doesn't make a difference. It wouldn't make a difference if the host opened the boxes or the contestants. It wouldn't make a difference if the host opened them and knew what was inside.
What would make a difference is if the host opens the boxes while deliberately avoiding opening the jackpot.
Lets reframe deal or no deal in a way that highlights the key part for me:
There are 25 boxes with sums from 1p to £1M in them. Nobody knows what's in them. You pick one randomly, and the others are placed in a pile. You now get to choose one of two things:
1. Keep the box you chose randomly
2. Take the highest value box out of the pile of 24
What if there were 24 boxes worth nothing and one with £1M in, does your strategy change?
What if there were 23 boxes with nothing and one with £1M in, does your strategy change?
