Quote from: Mishihari on August 18, 2022, 12:23:29 AM
l still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

I didn't believe it at first either.  For anyone that understand probability past a certain point, it's a magic trick of a reading problem.  The key is that the way it is described is to misdirect your training into thinking of it as a straight-forward probability exercise, when the real key is that the opening of the second door is done by someone who knows what is behind every door--including the one first selected.  If instead, the second door is open randomly, it does collapse into a simpler case, with the intuitive probability results.

Of course, in an RPG that complicates it more not less, because then there is the roleplaying element of ascertaining whether or not the party thinks the second pick is done by someone with or without that knowledge.

Years and years ago I used Monty Hall as part of sales training. Mishihari is not alone! In a room of 20, I'd typically get 1-2 people who completely dig in. What would *sometimes* work is expanding the number of doors when explaining why to switch.

Instead of there being three doors, one selection, and revealing one goat...

Assume there are 100 doors. You pick one which remains closed, a second door remains closed, and 98 open to reveal goats. Would you switch then? You freaking better. You only had a 1% chance of picking the right door originally.

And when they still protest...expand it to 1 million doors—the closed original pick, a second closed door, and 999,998 goats. Now your original pick was 1 in a million and the only other closed door is some rando out of 999,999 other doors. If they do not switch at this point, they are obtuse—deliberately or otherwise.

I have never used it in TTRPGs. This may be unfounded, but it feels a little quantum ogre to me. Still interested to hear how it goes or what other examples are given.

Quote from: Effete on August 18, 2022, 12:52:16 AM

Quote from: Mishihari on August 18, 2022, 12:23:29 AMl still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

More proof college is just a waste of time.

More like proof that even an intelligent person can be wrong.

I'd proffer other evidence as proof that college is a waste of time for many people. Conversely, I am living evidence that for some college is not a waste of time.

Quote from: Mishihari on August 18, 2022, 01:09:12 AM

Quote from: Effete on August 18, 2022, 12:52:16 AM

Quote from: Mishihari on August 18, 2022, 12:23:29 AMl still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

More proof college is just a waste of time.

Not really.  I certainly get paid a heck of a lot more than I would without multiple grad degrees.

I was jesting. But not really. College is not needed to make good money. But that wasn't my point either. My point was that even with multiple degrees in probability, the human brain can still be tricked by a fairly simple thought-exercise. I think FingerRod explained it best, but basically it goes like this: when you first selected a door your odds were 33.3% at being correct. When one if the other doors is opened, your brain wants to now change the odds to 50-50... but that's not accurate. The selected door RETAINS a 33.3% chance of being correct.

Increasing the number of doors makes it more obvious a switch should be made because the percentages change drastically (from 33.3/66.6 with three doors to 1/99 with 100 doors). That's what adds to the confusion: 33.3% still FEELS close to 50% so the person's brain assumes it is.

Quote from: Effete on August 18, 2022, 10:42:33 AM

Quote from: Mishihari on August 18, 2022, 01:09:12 AM

Quote from: Effete on August 18, 2022, 12:52:16 AM

Quote from: Mishihari on August 18, 2022, 12:23:29 AMl still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

More proof college is just a waste of time.

Not really.  I certainly get paid a heck of a lot more than I would without multiple grad degrees.

I was jesting. But not really. College is not needed to make good money. But that wasn't my point either. My point was that even with multiple degrees in probability, the human brain can still be tricked by a fairly simple thought-exercise. I think FingerRod explained it best, but basically it goes like this: when you first selected a door your odds were 33.3% at being correct. When one if the other doors is opened, your brain wants to now change the odds to 50-50... but that's not accurate. The selected door RETAINS a 33.3% chance of being correct.

Increasing the number of doors makes it more obvious a switch should be made because the percentages change drastically (from 33.3/66.6 with three doors to 1/99 with 100 doors). That's what adds to the confusion: 33.3% still FEELS close to 50% so the person's brain assumes it is.

I think that the "Simple Solution" on its Wikipedia page provides a good explanation (YMMV):
https://en.wikipedia.org/wiki/Monty_Hall_problem

I thought of a puzzle for such a dungeon, inspired by The Lady or the Tiger.

The PCs find a door with the following sign on it: "Within is a riddle. Three treasure chests beckon. One contains great wealth, the others certain death. The signs will guide you, but beware: not all are honest. Do you dare to enter?"

Within, the PCs find the the three chests, each with a sign. On the left chest the sign reads, "The middle chest contains the treasure." On the middle chest, "The left chest does not contain the treasure." On the right chest, "This chest does not contain the treasure." And above them is another sign: "Only one of these statements is true." In addition are the rotting corpses of another party, and the word "LIAR" is scrawled in blood on the wall.

If the PCs open the correct chest, they find a treasure (put whatever you want here), but if they open one of the others, a gate slams shut over the door and poison gas is released (save v. poison or die each round for 10 rounds). After 24 hours, the trap resets and the room becomes accessible again.

I'll leave the solution as an exercise, but leave a hint: yes, the solution is unique.

Quote from: Cat the Bounty Smuggler on August 18, 2022, 01:46:13 PM
Within, the PCs find the the three chests, each with a sign. On the left chest the sign reads, "The middle chest contains the treasure." On the middle chest, "The left chest does not contain the treasure." On the right chest, "This chest does not contain the treasure." And above them is another sign: "Only one of these statements is true." In addition are the rotting corpses of another party, and the word "LIAR" is scrawled in blood on the wall.

1) If the Left sign is true (treasure is in Middle chest), then the other signs are false and the treasure is ALSO in the Middle and Right chests. Therefore, the Left sign must be False.

2) If the Middle sign is true (treasure is not in Left chest), then the Left sign is false (treasure is not in middle) and the Right sign is false (treasure IS in the Right chest).

3) If the Right sign is true (treasure is not in Right chest), then the other signs are false and the treasure is in the Left chest.

Both 2 and 3 cannot be true at the same time.
The riddle lies with the fourth sign; the one on the wall. If one of the signs on a chest is true, the sign on the wall must be false, meaning that more than one sign must be True. That's what "LIAR" is referring to: the fourth sign. The sign on the wall is False, all the other signs are True, and the treasure is in the Middle chest.

Did I win?

Quote from: Effete on August 18, 2022, 02:38:44 PM

Quote from: Cat the Bounty Smuggler on August 18, 2022, 01:46:13 PM
Within, the PCs find the the three chests, each with a sign. On the left chest the sign reads, "The middle chest contains the treasure." On the middle chest, "The left chest does not contain the treasure." On the right chest, "This chest does not contain the treasure." And above them is another sign: "Only one of these statements is true." In addition are the rotting corpses of another party, and the word "LIAR" is scrawled in blood on the wall.

1) If the Left sign is true (treasure is in Middle chest), then the other signs are false and the treasure is ALSO in the Middle and Right chests. Therefore, the Left sign must be False.

2) If the Middle sign is true (treasure is not in Left chest), then the Left sign is false (treasure is not in middle) and the Right sign is false (treasure IS in the Right chest).

3) If the Right sign is true (treasure is not in Right chest), then the other signs are false and the treasure is in the Left chest.

Both 2 and 3 cannot be true at the same time.
The riddle lies with the fourth sign; the one on the wall. If one of the signs on a chest is true, the sign on the wall must be false, meaning that more than one sign must be True. That's what "LIAR" is referring to: the fourth sign. The sign on the wall is False, all the other signs are True, and the treasure is in the Middle chest.

Did I win?

Yep! Inside the middle chest you find a coupon for 1,000,000 fake internet points. Enjoy!

Seriously, though, well done.

Quote from: Cat the Bounty Smuggler on August 18, 2022, 06:33:30 PM

Quote from: Effete on August 18, 2022, 02:38:44 PM
Did I win?

Yep! Inside the middle chest you find a coupon for 1,000,000 fake internet points. Enjoy!

Seriously, though, well done.

Awesome!

I'm taking them to DTRPG to buy some fake games.

Quote from: Mishihari on August 18, 2022, 12:23:29 AM
I stil

Quote from: Cat the Bounty Smuggler on August 14, 2022, 05:16:43 PM
The Monty Hall Problem is a counter-intuitive result in probability. Long story short, given three doors one with the Big Prize and two with goats, you pick one. Hall reveals a goat behind one of the other two and gives you an opportunity to stick with the door you've already chosen or switch to the other unopened door.

It seems like switching should make no difference, that there's a 50/50 chance of getting the Big Prize, but in fact switching increases your chances from 1/3 to 2/3, so you should always switch doors.

This is easy to prove but it's difficult to get people to believe it because it's so absurdly counter-intuitive. I know it took me a while to accept it.

ETA: Let me try to prove it. When you first pick one of the three doors, there's an equal chance that any of them can be the Big Prize, so the chance you picked correctly in 1-in-3, and the chance you picked incorrectly in 2-in-3. Hall opens one of the latter two doors and reveals a goat (there's a goat behind at least one of them, so he can always do this). The key is that this hasn't changed the probabilities, it's simply eliminated one of the wrong options. So at this point the probability you chose correctly is still 1-in-3, and the probability you were wrong is still 2-in-3, but now there's only one wrong option instead of two. So by switching, you get the full 2-in-3 probability of winning the Big Prize.

If that still doesn't do it for you, consider an alternate scenario. Suppose after picking a door, Hall offers you to chance to open both of the other doors, and you get the Big Prize if it's behind either. Then obviously you should do it, because opening two doors is obviously better than one. Well, in the original scenario, by switching you are effectively opening both of the other two doors.

l still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

For what it is worth when I was younger I have actually seen this problem played out as an experiment with a sample of a few hundred people in a gymnasium and sure enough the results match the solution.

I think as Steve said the problem is partially a word puzzle and also plays on humans being quite bad (or at least easily tricked) when it comes to assigning cause and effect and control over an environment.

In other words you made a choice of three doors. You have a 1 in 3 chance of getting it Right. You now know one of the doors was Wrong. But nothing else about your initial choice has changed despite the visual cues to the contrary.

You had a 1 in 3 chance of having made the Right choice before and you have a 1 in 3 chance of having made the right choice afterwards.

Given these facts and the choice to change, logic tells you the other door must have a 2/3rds chance of being Right.

Incidentally in terms rping Monty Haul dungeons do have a place as a campaign finale and giving the PCs a reason to retire!

I just recalled a great example, the Hidden Shrine of Tamoachan module for 1E.  As I recall it was entirely a puzzle dungeon without a single monster in it.

Quote from: Visitor Q on August 19, 2022, 06:16:31 AM

Quote from: Mishihari on August 18, 2022, 12:23:29 AM
I stil

Quote from: Cat the Bounty Smuggler on August 14, 2022, 05:16:43 PM
The Monty Hall Problem is a counter-intuitive result in probability. Long story short, given three doors one with the Big Prize and two with goats, you pick one. Hall reveals a goat behind one of the other two and gives you an opportunity to stick with the door you've already chosen or switch to the other unopened door.

It seems like switching should make no difference, that there's a 50/50 chance of getting the Big Prize, but in fact switching increases your chances from 1/3 to 2/3, so you should always switch doors.

This is easy to prove but it's difficult to get people to believe it because it's so absurdly counter-intuitive. I know it took me a while to accept it.

ETA: Let me try to prove it. When you first pick one of the three doors, there's an equal chance that any of them can be the Big Prize, so the chance you picked correctly in 1-in-3, and the chance you picked incorrectly in 2-in-3. Hall opens one of the latter two doors and reveals a goat (there's a goat behind at least one of them, so he can always do this). The key is that this hasn't changed the probabilities, it's simply eliminated one of the wrong options. So at this point the probability you chose correctly is still 1-in-3, and the probability you were wrong is still 2-in-3, but now there's only one wrong option instead of two. So by switching, you get the full 2-in-3 probability of winning the Big Prize.

If that still doesn't do it for you, consider an alternate scenario. Suppose after picking a door, Hall offers you to chance to open both of the other doors, and you get the Big Prize if it's behind either. Then obviously you should do it, because opening two doors is obviously better than one. Well, in the original scenario, by switching you are effectively opening both of the other two doors.

l still don't believe it, and I've had several graduate level courses in probability.  I figure that is something is that massively counterintuitive then there's something wrong with the proof.  Just because I'm not smart enough to point out the flaw doesn't mean it's not there.

For what it is worth when I was younger I have actually seen this problem played out as an experiment with a sample of a few hundred people in a gymnasium and sure enough the results match the solution.

I think as Steve said the problem is partially a word puzzle and also plays on humans being quite bad (or at least easily tricked) when it comes to assigning cause and effect and control over an environment.

In other words you made a choice of three doors. You have a 1 in 3 chance of getting it Right. You now know one of the doors was Wrong. But nothing else about your initial choice has changed despite the visual cues to the contrary.

You had a 1 in 3 chance of having made the Right choice before and you have a 1 in 3 chance of having made the right choice afterwards.

Given these facts and the choice to change, logic tells you the other door must have a 2/3rds chance of being Right.

Incidentally in terms rping Monty Haul dungeons do have a place as a campaign finale and giving the PCs a reason to retire!

I couldn't wrap my head around it until I realized the wrong doors are paired.  Any time you choose a wrong door you are choosing both wrong doors. 

Or another way of putting it is the realisation that Monty revealing there was a goat behind one of the doors doesn't actually give you any more information than you already knew.

Pick any two doors and there is always at least one goat. Knowing which specific door doesn't matter. That's just revealed for dramatic effect.