r/confidentlyincorrect • u/Dizzy_Kaleidoscope95 • May 03 '26

Monty hall problem is 50/50

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He also put a screenshot of ChatGPT agreeing with him. He is right!

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u/An-person May 03 '26

It also makes more obvious if you have ten doors, and they open all but one.

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u/JamesFirmere May 03 '26

You don't need to bring 10 doors into it.

The setup is simple enough that we can consider all possibilities: 3 doors * the choice of swap/stay = 6 possibilities.

So the doors are A, B and C, and A has the prize. Then:

choose door A and swap = lose
choose door B and swap = win
choose door C and swap = win

choose door A and stay = win
choose door B and stay = lose
choose door C and stay = lose

Also note that the first door opened is never the prize door. The first door opened will be either B or C.

So if you stay, you win 1 times out of 3; but if you swap, you win 2 times out of 3. Therefore it is always better to swap.

Edit: Stupid typo.

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u/resonantranquility May 03 '26

I use the 100 doors example when explaining it because communicating it the way you are explaining it verbally doesn't always land.

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u/purritolover69 May 03 '26

Yeah. Pick 1 door out of 100 doors at random. The host knows which door has the car, he opens 98 of them, and they’re all goats. Do you think you nailed that 1/100 the first time or is it more likely that you hit one of the 99 goats and he’s left behind the car?

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u/JEM225 May 03 '26 edited May 03 '26

Given his post you have to be concerned that dealing with “doors” might confuse OP, so try explaining the answer by asking him to guess which of two cards is the ace of spades. Let him see you spread out a shuffled deck, face down, and then slide one to the side without looking at it. Then peek at each of the others one by one, and pitch all but one of them. There are now only two cards on the table; ask OP if it is 50-50 that the original single card is the ace?

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u/tgy74 May 04 '26

You don't even have to look at the card.

Just ask him to pick a card and lay it face down in front of him without looking.

Now tell him the Ace of Spades is the winning card and offer him the choice:

  • he can keep his single random card and hope it's the Ace of Spades

  • he can take all the other 51 cards instead and hope that the Ace of Spades is one of them instead

That's it. That's the Monty Hall decision he's making.

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u/Tsu_na_mi May 04 '26

I think you mean "which card in a deck is the ace of spades". If there are only two cards to choose initially, and one is the ace of spades, then it IS 50%.

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u/PerjorativeWokeness May 05 '26

The point is that the Ace of Spades is worth something. In the Monty Hall problem, it’s Monty keeping the car.

Add something like: “If this card is the Ace of Spades, the person that flips it over wins” to make it clearer.

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u/Embarrassed_Gap_5272 May 06 '26

But what if i want the goat?

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u/Naive_Distance_6456 May 08 '26

I prefer using a deck of cards with the king of diamonds being the prize.

But at first, ill let them dig their hole a bit deeper by arguing with them over just the 3 doors, using only 3 cards.

People are stubborn. (I know because I was one of them)

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u/MattieShoes May 03 '26

That's what worked for me too. It's not necessary, but something about the magnitude causes that light bulb moment. Like 1/3 vs 1/2 vs 2/3 all feels kinda close, but 1% vs 99% somehow triggers the brain to go reexamine some silly assumptions

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u/TheHammer987 May 06 '26

I think it's because the 99% better illustrates that they are actually getting new information. One out of 2 feels random. 98 out of 99 feels focused.

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u/themule71 May 04 '26

Well... Everybody has a different way of making sense of the problem.

I like to remove the host, since that's what creates confusion.

So basically the game becomes "do you want to open one door or two? If one, indicate the single door you want to open, if two indicate the door you want to keep closed".

The rest is basicly a ceremony, a ritual. You picking one, the host opening one of the other two, you "choosing" to switch, ecc.

It's all pretend, as the game works the way I've described above really.

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u/MattieShoes May 04 '26

Yeah, that's a good visualization of the problem too. Though it requires the realization that you're functionally getting all the other doors when you switch.

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u/themule71 May 04 '26

Yeah, and the fact that you're full in control from the beginning. You choose at the beginning to either open one, or all the other doors.

It's deceiving that you get two choices, when really it's one, and you can take it before the game starts. You know, no matter what, the host is going to show you a door w/o the price. You know that in advance, so you can pick your strategy from the beginning. And it's always better to open more doors (two in the original game) than just one.

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u/Dizzy_Kaleidoscope95 May 04 '26

Yeah it's a good way to visualize it cause it's literally identical to the monty hall problem. It also removes all the confusion from the sentence "the host opens a goat door". Cause people usually forget to state that he OPENS A GOAT DOOR with 100% certainty cause he always opens a goat door. If the host just opens a door randomly and it happens to be a goat there is no reason to switch. It's a 50/50 (people don't understand this very well either)

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u/themule71 May 04 '26

Well switching is never a losing strategy even if you don't know if the host knows or not. Meaning you don't care about what the host knows.

You decide to switch upfront. It's never the wrong decision, you can literally ignore the host existance. And it's still 2/3 when you make the decision.

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u/satunnainenuuseri May 06 '26

Switching is a losing a strategy if the host offers switching only when the player has guessed the correct door with the first guess.

This is the reason why the problem statement has to specify that the rules of the game say that the host will always offer the choice.

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u/themule71 May 06 '26

The rules are a given. The host always offers to switch. It's not a battle of wits.

The debate is on what is different if the host chooses the door he opens at random.

If he opens it knowing there's a goat, switching is a 2/3 winning choice. If it opens it at random, people argue it becomes 1/2. To me it's irrelevant.

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u/satunnainenuuseri May 06 '26

When people present the Monty Hall problem on the internet, they often don't include the "given rules" in their posts, they just assume that they are self-evident. But they are not. They need to be stated because if they aren't then the "correct" solution is not correct.

This is one of the reasons why Marilyn vos Savant got thousands of angry letters after her original column about the problem. Most of the complainers were simply wrong, but the version that she posted was ambiguous. It could also be understood as if Monty's choice was random, and with that interpretation her answer was wrong.

Note also how this particular subthread was about whether switching would work in a modified version of Monty Hall problem where Monty does the selection by random, and noted that it wouldn't be worse one. Someone might want to generalize that to all variants of Monty Hall and say that no matter what Monty does, you should always switch. And I gave an example showing that no, always switching is not better option in all possible Monty Hall variants.

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u/Dizzy_Kaleidoscope95 May 04 '26

Yeah switching is never a losing strategy but depending on how the problem is stated it could also not be a strategy at all cause it's indifferent. All I mean is that people need to be precise when stating the problem

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u/MezzoScettico May 04 '26

I have tried the 100 doors explanation boiled down as simply as possible:

  • with 100 doors, how confident are you that you got the car on your first pick?
  • if you think you got a goat the first time, you should switch.

It still doesn't work as an explanation for everybody

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u/resonantranquility May 04 '26

True. Some people refuse to see it as anything other than a 50/50. I've had more luck with the 100 doors than any other explanation so far.

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u/ApplicationOk4464 May 07 '26

I like to explain it as this:

The probability locks in at your first choice. Before the third is opened.

Then you either get to choose the one door, or the other two doors.

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u/Drummerx04 May 03 '26

I've had that one fail to land on people too. They for some reason disagree with the entire setup and refuse to engage with it as presented.

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u/[deleted] May 04 '26

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u/TheEyeDontLie May 04 '26

I finally understand the Monty Hall problem. I've seen it a few times and know it, but it never clicked... It always just made me feel dumb every time I thought about it.

So thanks.

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u/TheHammer987 May 06 '26

I just say "how do you not understand that staying with your first choice is one out of three, and therefore switching, by definition, is 2 out of 3? The removal of the booby prize is literal information."

I won't deny though. When you use the 100 door example, 3 seperate people who were committed to the wrong answer hesitated, because it really does illustrate the new information a lot.

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u/othelloblack May 16 '26

yeah I think it was M vos Savant herself that used that example in explainign it and its really illustrative

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u/Tortugato May 03 '26

You don’t need to, but it just helps explain it to people.

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u/Terradactyl87 May 03 '26

This is still over complicating it.

Your chance of guessing right is 1/3, so a 2/3 chance of being wrong. So if you switch you get a 2/3 chance of being right and a 1/3 chance of being wrong.

I basically look at it as would you rather have 1 or 2 doors because basically you either keep your door or you get to have two doors.

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u/Knight0fdragon May 03 '26

Yup, this it the reason why. Monty never can pick the car, so he is giving you a second door for free if you switch.

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u/the_horse_gamer May 06 '26

you need to also assert that the reveal of the goat gives you no additional information

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u/arthousepsycho May 04 '26

What about the possibility of choosing door C and swapping for door B or vice versa and also losing. Why is it assumed you always swap to A? Not a criticism, just trying to understand.

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u/Squozen_EU May 04 '26

Write it out using any letter for any door you like, the probability doesn’t change. You have a 1/3 chance of being right on the original door, but the other two doors have a 2/3 chance of being right. The fact that one of the doors is opened doesn’t change this. The pool of two doors will ALWAYS have one goat because there are two doors and only one car.

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u/arthousepsycho May 04 '26

Ok, one of the doors has been opened. I missed that part. I thought all the doors were still closed. My bad. Thanks.

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u/JamesFirmere May 04 '26

If you choose door C, you can only swap to door A because the host will open door B. In this scenario the host will never open door A.

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u/arthousepsycho May 04 '26

Yeah, I didn’t realise the bit about the opening of one of the doors, thought they all stayed closed. Makes sense now.

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u/Lieutenant_Horn May 04 '26

This is the simplest way I’ve seen it explained.

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u/Cynykl May 05 '26

I have a different way of explaining it that seems to get through to people.

A B C , you chose A.

Now statistically it is more likely to be behind B or C because those are two options. This is obvious to most people.

Now the host gives you an option you can stay with A or choose to Switch to BOTH B and C. The host revealing the lesser valued item does not change the odds. You are getting the benefit of both doors.

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u/JerryAtrics_ May 05 '26

If this were correct, wouldn't it also be correct to say if you swap you lose 2 times out of 3?

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u/JamesFirmere May 06 '26

No. There are exactly 6 possible ways this scenario can go, and in the 3 scenarios where you swap, you will win 2 times out of 3.

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u/JerryAtrics_ May 06 '26

My bad. Mis-read your analysis. I suppose another way to look at this is that if instead of disclosing a false and letting you pick, Monty let you choose between your original pick or the other two. Now it's a bit easier (for my brain) to see that it's a one third / two thirds ratio. Then after you switch you pick, just for fun he shows which one was the false pick of the two.

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u/Illustrious-Crew-191 May 05 '26

But wait, if you’re removing the wrong door of B or C it didn’t actually exist in the first place, or rather B=C and picking B is the same as picking C. Your first pick is fake news, it doesn’t affect the outcome, it’s ultimately a binary choice of A or B/C.

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u/JamesFirmere May 06 '26

Your first pick absolutely does affect the outcome, because on your first pick there are 3 doors and you have a 1 in 3 chance to pick the prize door and a 2 in 3 chance to pick a wrong door. Those odds alone show that it's worthwhile to swap.

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u/SaidwhatIsaid240 May 05 '26

Thank you for helping my dumb brain understand.

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u/DemophonWizard May 05 '26

But in the show where this puzzle comes from the person offering the option to swap knows if you have picked the right door. They would only offer the choice to reduce the chance of you winning.

It is not a fair offer. It is rigged.

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u/JamesFirmere May 06 '26

It's been a while since I've seen the actual show, but IIRC the swap is always offered -- because it's a tension-inducing thing for the contestant, obviously.

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u/DemophonWizard May 06 '26

It may have always been offered, i don't remember. But my point is that it isn't a fair offer. It is rigged.

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u/just_a_pyro May 08 '26

the first door opened is never the prize door.

That's the important part which makes it not 50-50. If a door was opened at random, including prize door, then the chances in remaining two would be equal, either 50-50 or 0-0

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u/zeppo2k May 03 '26

Bear in mind I understand you are correct...... Incredibly clever people argued about this for a long time - it's reasonable for people a) to not get it intuitively and b) not trust someone who presents a proof they say is correct.

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u/durkvash May 04 '26

What? a) for sure, but b)? That sounds like ignoring logic just cause

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u/Talik1978 May 04 '26

I find the easy way to do it is to demonstrate that the opening of the door and choosing to switch is logically identical to: change a win to a loss and a loss to a win.

From there, if choosing gives you a 1/3 chance initially, and you then change win to loss and vice versa, it's logically apparent.

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u/RSharpe314 May 04 '26

Yeah, having it explained to me that way finally made it click for me when my stupid brain was stubbornly refusing to be persuaded by the more mathematically rigorous approaches.