@DavidCoffron So this is the first chance I've had to actually look at this. Gonna take a pass at some stats on this maybe this week.

Been on vacation for a week. =P

I ended up nerd-sniping myself over the weekend thinking about the "Drunk Monty Hall Problem".

It's similar to the classical Monty Hall Problem, but in this variant, Monty Hall arrives drunk onto the set, and has completely forgotten which door contains the money, and which doors contain the goats.

So when you pick a door, he randomly opens one of the two unpicked doors, and he may or may not inadvertently reveal the money.

But, for the sake of argument, let's say he opens a door containing a goat, and gets to keep his job, and then (per the classical problem) offers you a choice to switch to the remaining unopened door. Is it advantageous for you, the contestant, to switch doors?

.... Speaking as someone who took to the original problem relatively well, this one stumped me for a little while, and I had to whip up a simulation to prove what the actual correct answer was.

Maybe an oversimplification but intuitively I'd say this:

Meaning switching or not is the same in this case?

Also I have spent nearly 20 minutes on this when I should be working :)

@Sdjz Yup!

So weird, my first instinct was that it had to be the same as the original as the contestant doesn't know that the host picked randomly

But then you are ensuring that the random choice never picks gold

@Sdjz Yeah, the trick is realizing that the odds that Monty reveals a goat, given you picked a goat originally, are only half as high as in the regular problem. You normally have a 2/3rds chance of having not picked the money originally, but of that 2/3rds scenario, half your chances of winning are being screwed because of Monty's unprofessionalism.

Here's the puzzle on M.SE:

I really like the general case provided by graham Kemp. You can plug in how likely it is that Monty reveals the goat.

@Xirema What about Monty Hall with 4 doors?

@DavidCoffron Depends. Does Monty reveal two doors, or only one? And is he drunk or sober?

@Xirema Reveals 1. 100% chance that he reveals a goat

Then switching should yield the money 3/8ths of the time, as opposed to sticking with your original choice which yields the money 1/4th of the time.

3/8? That's not what I got...

I might be wrong though; one sec

@DavidCoffron Well, bear in mind you still have to choose one of the two remaining doors, the collective probability for them both is 3/4, but split across two doors.

@Xirema Ohhhh. I was thinking if you could still choose the door that Monty revealed; obviously you wouldn't do that. Duh

I get 1/4 since I was saying the 3/4 split across 3 doors. That's not how math works

Is there a number of doors where it is no longer beneficial to switch?

It's (NumberOfGoats/NumberofDoors)/(NumberofUnrevealedUnchosenDoors)

@DavidCoffron No, but the benefits to switching become extremely minimal after even a small number of additional doors.

The only way to make it suddenly non-beneficial to switch is if Monty isn't playing fair (per the original agreement, i.e. always reveals a goat, always offers to switch)

Like, you could imagine a game where Monty always reveals the money, if it's an unpicked door. In that scenario, switching will always lose the game, because the only time he ever reveals a goat is if you picked the money originally.

@Xirema Yeah, ((X-1)/X)/(X-2)>(1/X), but by linearly less as more doors get added

What if Monty Hall but behind the door are the friends we made along the way?

@Yuuki Do the friends get to shout from behind the door? How thick are the doors?

@Yuuki Depends. Does he put one friend behind each door? Is this a Sophie's Choice situation?

@Xirema You are better off staying if the number of doors is between 0 and 1 :P

What if Monty Hall shows up drunk to the set, but the producer is a Time Lord who refuses to be humiliated, so after the contestant chooses a door, the producer continuously loops time until Monty correctly opens a door to reveal a goat?

@Xirema Are we talking old series, new series, or that one episode with the paradox-sucking monsters

@DavidCoffron As someone who has watched exactly one episode of Doctor Who: yes.

@Xirema Then the answer is, timey-wimey stuff happens

Okay 4-doors Monty Hall, but he reveals 1 goat, asks if you want to switch, then reveals another goat (which can be your original choice if you switched) and asks if you want to switch. Is it the same as if he revealed 2 doors right away?

@DavidCoffron Ooof.

Yeah, I'm getting 5/8 but I'm unsure

(as opposed to 3/4 for revealing 2 doors right away)

Switching once and staying is 3/8 like the one-reveal 4-doors. Staying is obviously 1/4. Switching again means the 5/8 chance you are presently on a sheep gets compounded into the 1 remaining door to switch to. Right?

And then the fun part would be explaining why its different than if he revealed 2 sheep right away.

I think that's all correct, and it's different because when revealing 2 right away he can't choose the original one you chose. Just like in the original problem, if Monty opens a door before you have chosen anything then it's just 50/50

@Sdjz Ah. That makes much more sense than I thought it would

I think its interesting that if you switch away, and he shows that the other one you could've switched to is a sheep, its best to switch back.

CDDD -> CDDS -> DCDS -> DCSS -> CDSS

Isn't it 5/8 on the one you didn't pick on the second step regardless?

Because like you said, switching on the second step hits gold 3/8 of the time so the other remaining door must always be 5/8

@Sdjz There are three available doors at that point. It isn't until Monty reveals that one of those doors is a sheep that 5/8 is available in that other door

So it's 3/8|3/8|2/8

(where the 2/8 is the one you switched away from when Monty revealed the first sheep)

I think we are agreeing

So, yeah no matter which one he shows is a sheep, the 5/8 is there

(since basically the chance that one was a prize moves to the available door)

@Sdjz oooh. You were saying that it doesn't matter which one he reveals, you have 5/8 if you switch. Yeah, when I was talking about the interesting factor I meant just interesting from an intuition sense; not that it would have a different probability. In my intuition, I would think its best to stay because I have the confirmation bias acting that I must have switched on to the prize if he's showing me that the other switch was the sheep.

@DavidCoffron Yeah I didn't understand that that was what you meant at first, only after rereading

@Sdjz The classic case of agreement during perceived disagreement

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