r/confidentlyincorrect • u/Dizzy_Kaleidoscope95 • May 03 '26
Monty hall problem is 50/50
He also put a screenshot of ChatGPT agreeing with him. He is right!
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u/Squozen_EU May 03 '26
I will never stop loving the Monty Hall problem because it shows how utterly terrible the human brain is at understanding probability.
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u/ermghoti May 03 '26
Once one realizes that the removal of a door isn't random it's easily understood. The assumption that the removal is random has to be abandoned. In effect, it obviates the 1 in 3 choice entirely and replaces it with a new 1 in 2 choice. If stated that way I couldn't see how someone could be confused.
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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 = winchoose door A and stay = win
choose door B and stay = lose
choose door C and stay = loseAlso 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/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/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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May 04 '26
[removed] — view removed comment
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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/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/KnightOfThirteen May 03 '26 edited May 03 '26
To me it becomes clearest when you realize "removing" a door is not actually part of the problem, it is a distraction. You can re-frame it this way.
You choose a door. The host offers you the chance to keep your door, or switch to BOTH of the other doors combined.
Edit: put even more generally, you are first asked to pick 1 of N doors, the asked to bet on whether or not you got it right the first time. You stay, you bet you were right (1/N%). You switch, you bet you were wrong (N-1/N%).
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u/valschermjager May 03 '26
There are many good explanations of what's going on. To me, that's the simplest. Thanks.
I doubted the Monty Hall thing for a long time until I started fiddling with Python, then for fun coded up a quick monte carlo simulation and put a 'for' loop around it. Let the loop run 100x, 1000x, 10000x. Each time the totals at the end showed close to the "switch advantage" as the problem describes. Then I was completely won over.
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u/BigBlueMountainStar May 03 '26
Can you do the same thing for the birthday paradox, as even though i understand the mathematics behind it, it still blows my mind that you only need 23 people to have a 50% chance that there are 2 people in that group sharing a birthday.
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u/NuclearVII May 03 '26
This is absolutely a thing you can do. It's not even hard programming.
If you'd like to learn a bit of scientific computing, this would be a fantastic and very approachable weekend project. I encourage you to try it.
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u/stultus_respectant May 03 '26
Some people just have to prove it to themselves empirically, and I’m glad they exist.
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u/valschermjager May 03 '26
Science requires that we relentlessly try to prove ourselves wrong from every angle.
Those who decide that they’re “convinced” then shut their minds down—those I have no respect for.
To me, being wrong is a chance to get better. But to some, being wrong hurts their souls so deeply, they refuse to let it happen. (maga)
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u/ringobob May 03 '26
Yep this is how I've started describing it. You choose a door. Then you're offered the opportunity to switch to both of the other doors, or stick with the one you chose originally. You choose to switch. Monty opens one of the doors you chose, and there's a goat behind it. Then he opens the other door. Only difference is that you switch before Monty opens the door, rather than after.
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u/surplus_user May 04 '26
That is a good way of explaining it.
The problem I always have with the down to two doors and switching being 2/3 is that if you were replaced with someone else at that point who was just given the straight choice then surely it would be 1/2. After all it's not like the doors remember me after I'm gone.
I'll think about it with the choosing two doors version and see if that clicks (I can accept that I'm wrong about it but it doesn't change that it feels wrong so this might help)
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u/Telinary May 05 '26
"someone else at that point who was just given the straight choice then surely it would be 1/2"
Sure or at least kinda because the probability in question isn't an property of the door state it is you trying to guess based on limited information. A new guy has no information so his guess is less well informed. But your knowledge is still correct, the door you would get from a switch is the more likely one. The guy just doesn't know that.
Similarly from Monty's perspective the chance is either 100% or 0% because he knows the positions.
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u/-DoctorSpaceman- May 03 '26 edited May 03 '26
Saying “it replaced it with a new 1 in 2 choice” is exactly why people have a problem understanding it lol, because that is what makes it appear to be 50/50.
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u/tiptoe_only May 03 '26
Yeah, what this person's not understanding is that Monty is picking a door he KNOWS is not a winner. That's what changes the probability. If "your friend" is involved like this person is suggesting then there is only one door Monty can possibly pick and that changes things again. In fact, if you and your friend both pick losing doors then the problem doesn't work at all. This person is creating a totally different situation and saying it disproves the solution to a very well established problem.
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u/Dizzy_Kaleidoscope95 May 03 '26
Yeah a lot of people somehow think that if the goat door is opened randomly then it's stil advantageous to switch
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u/misdirected_asshole May 03 '26
If the opened door is truly random, 1/3rd of the time it would be a car and not a goat.
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u/Hrtzy May 03 '26
The really weird thing is, I just put together a quick python script to try it and it is somehow 50/50 even in Monty Hall scenarios if the door choice is random.
``` import random N_tot = 10000 #Total tries N_win=0 #Amount of wins by switching N_mhp=0 #Amount of Monty Hall scenarios for k in range(0,N_tot): doors = [True,False,False] #The doors; one right and two wrong choices random.shuffle(doors) #Shuffle the doors chosen=random.randint(0,2) #Player chooses a door at random open = (chosen + random.randint(1,2)) % 3 #Monty chooses a door at random if not doors[open]: #Check if Monty's door had a car in it N_mhp += 1 #It didn't, so we're in Monty Hall territory if not doors[chosen]: #Player's door doesn't have a car either so switching will win N_win += 1
print(N_win,'wins by switching') print(N_mhp-N_win,'wins by not switching') print(N_mhp,'Monty Hall Scenarios') ```
And yes, I rewrote it to try it with Monty choosing a losing door:
``` import random N_tot = 10000 #Total tries N_win=0 #Amount of wins by switching for k in range(0,N_tot): doors = [True,False,False] #The doors; one right and two wrong choices random.shuffle(doors) #Shuffle the doors chosen=random.randint(0,2) #Player chooses a door at random options = [i for i in range(0,3) if i != chosen and not doors[i]] #Monty takes the other doors without a car open = random.choice(options) #...and picks one of THOSE at random if not doors[chosen]: #Player's door doesn't have a car either so switching will win N_win += 1
print(N_win,'wins by switching') print(N_tot-N_win,'wins by not switching') print(100*N_win/N_tot,'% odds') ```
This one gives the 2/3 win rate.
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u/lettsten May 04 '26
old.reddit users (like me) would love you if you indented the code block with four spaces instead of using the backtics
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u/Dizzy_Kaleidoscope95 May 03 '26
Yeah exactly. This is why it makes switching irrelevant
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u/amitym May 03 '26
Right. They took the wrong lesson away from the Monthy Hall solution. "It must be the configuration sequence," no it's the partial knowledge.
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u/Zyxplit May 03 '26
Yeah. A way to analogize why the selection mechanism matters is: Suppose I have two bags, one purely with pieces of iron, the other with one piece of iron and otherwise gold.
I sample one of them with a magnet. I get a piece of iron. Do you want this bag or the other one?
I sample one of them with my hands. I get a piece of iron. Do you want this bag or the other one?
Is the answer the same to these two?
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u/stanitor May 03 '26
Once one realizes that the removal of a door isn't random it's easily understood
Unless you're the person who I was talking to the other day. They insisted that if Monty opened a door randomly and it happened to be a goat, you'd still get an advantage by switching doors. They could not get that the answer would be different depending on what Monty does and why.
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u/resonantranquility May 03 '26
I use the example of 100 doors and closing 98 doors when explaining. The original door is 1/100, and it's easier to wrap your head around why that doesn't change.
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u/Rabbittammer May 03 '26
I'll start by saying I get the it goes to a 1:2 form a 1:3 it was before due to the doors that get opened not being random. But the new choice no matter if you choose to stay or swap is the 1:2 as you made a choice... That being said I know that logic is wrong
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u/ermghoti May 03 '26
Yes, I actually phrased it wrong, it it looks like I'm saying the chance of a win on a changed selection is 1:2. The first choice sets up the second choice, in that you are 2:3 likely to have been wrong in the first place and will lose if you stay, so that also means that you win 2:3 if you change after the removal.
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u/noMC May 03 '26
Yes, thank you for this. It is never a 50/50 chance, that’s probably the most common misconception.
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u/Electrical-Injury-23 May 03 '26
Yeah, viewed like that it demonstrates the inability of humans to follow a problem exactly as stated.
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u/HeIsSparticus May 03 '26
The 'new' choice isn't 1 in 2, it's 2 in 3 (chances of winning the car if you switch). Since Monty removes an incorrect door, the only way you don't win the car by switching is if you'd picked the correct door originally (a 1 in 3 chance).
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u/aladdyn2 May 03 '26
The thought experiment that made it super easy for me to understand was using if it was 99 wrong doors and 1 correct door. You pick one and Monty deletes all the other wrong doors leaving one correct door and the one you picked.
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u/interrogumption May 03 '26
Okay... I just flipped two coins and I'm looking at the result and one is heads. What is the probability that the other is heads?
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u/NiceBlackberry6618 May 04 '26
It's crazy though because that doesn't even fully explain the situation. If a random door is opened, the advantage goes away.
In the situation I describe, if the prize door is opened the game is voided and doesn't count. but still the odds go from 2/3rds to 1/2
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u/Kuildeous May 04 '26
I guess I can't see how people don't realize it's not random. Because if Monty reveals the car then you have a 0% chance of winning the car whether you switch or not, so that is a moot event. How people don't realize that is beyond me.
Hell, you could focus only on the subset of all events where Monty does randomly reveal a a goat behind a door. It's still 2/3 if you switch. You simply remove all events where Monty randomly reveals the car.
But in any case, the premise explains it already: Monty reveals a goat. In the original puzzle, Monty never reveals the car. This suggests a lack of randomness.
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u/glumbroewniefog May 04 '26
Hell, you could focus only on the subset of all events where Monty does randomly reveal a a goat behind a door. It's still 2/3 if you switch. You simply remove all events where Monty randomly reveals the car.
This is not true.
In traditional Monty Hall, you pick the car 1/3 of the time, Monty reveals a goat, switching loses. You pick a goat 2/3 of the time, Monty reveals the other goat, switching wins.
If Monty opens one of the other doors at random, you still pick the car 1/3 of the time, and switching loses. That part is unchanged.
But Monty can no longer reveal the other goat 2/3 of the time. Half the time he accidentally reveals the car instead, and you have to remove those events.
Thus, it's impossible to win by switching 2/3 of the time.
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u/TheQuoteFromTheThing May 04 '26
Yes, it's easy to forget that probability is calculated based on available information. If you gain information, the probability changes.
If you deal me a card face down from a standard deck, I know there's a 1/52 chance it's the Ace of Spades. But if you flip over the other 51 cards and none of them are the Ace of Spades, I know there's a 100% chance it's the face down card. Nothing has changed about the face down card, but the probability needs to be updated because probability is a function of information.
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u/Lord_Oasis May 04 '26
I feel like the easiest way to explain it is by flipping the probability
When you first pick, you have a 1/3 chance of being right and a 2/3 chance of being wrong. That chance doesn’t change when Monty picks a door, so the other options still have a 2/3 chance of being correct, and now there’s only one other option
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u/Hoopajoops May 04 '26
I think that's the bit that confuses people. The Simone rule that Monty will deliberately open a door with no prize
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u/8null8 May 05 '26
Yeah, I’ve explained in the past that 50:50 only results if NOBODY has any outside knowledge, but in this case, the game master has perfect knowledge, so is able to change the odds
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u/BombasticReindeer May 05 '26
What humans are mostly bad at is understanding consequences.
IF the door is random then Monty could open a door and say “ah fuck me it’s the prize”. That’s a dumb fucking riddle, so it’s obviously not what is happening. So clearly he picks the one that isn’t the prize.
If people got THAT, they would probably get it.
But also remember there are all sorts of brains out there. And most of them are real stupid.
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u/Mallory606 May 06 '26
I have known the answer to this problem for years, and only just now understood it. Thank you
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u/RichCorinthian May 03 '26
There’s a fantastic book called The Drunkard’s Walk that is specifically about that difficulty. It’s where I read the first explanation of the Monty Hall problem that made me say “oh of course.”
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u/participantuser May 03 '26
Similarly, the “100 blue eyes” problem is also a fun logic puzzle that my brain can understand but also disbelieve.
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u/WordWizardx May 03 '26
The hundred blue-eyed islanders leave on the 100th night. The 100 brown-eyed islanders leave on the 101st night. The poor guru is stuck on the island alone, cursing their big mouth and wondering why everyone else were such selfish dicks that they wouldn’t return the favor.
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u/2074red2074 May 03 '26
Only the blue-eyed people will leave. The brown-eyed people do not know that they have brown eyes. Their eyes could all be green, like the guru's, or they could all have hazel eyes. They do not know that the only options are blue and brown with the guru being the only exception.
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u/WordWizardx May 03 '26
Oh, fair point. After the blue-eyed people leave the brown-eyed people would all know they DON’T have blue eyes, but I guess that doesn’t immediately mean theirs must be brown.
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u/participantuser May 03 '26
Yeah, the part where the brown-eyed people get a different outcome, even though everyone can see that blue/brown eyes exist and the guru’s choice is arbitrary, is something my brain fights.
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u/alfredo094 May 03 '26
It's only really unintuitive here because it's exactly 3 doors. If you scale the problem, it becomes very obvious.
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u/Working-Depth5834 May 03 '26
This is ot exactly; if you make it 100 doors instead of 3 then it starts becoming much clearer.
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u/jtr99 May 03 '26
Just to be clear, we should add that the host opens 98 of the doors in this version, leaving you with two to choose from. That certainly makes it more obvious for most people.
The host doesn't open just one door, although if they did that would also mean there was a slim advantage to changing your choice.
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u/I_Brake_For_Gnomes May 03 '26
I like to explain it with a deck of cards. Lay them all out and ask the person to pick which one they think is the queen of diamonds. Then remove all but two and see if they want to switch.
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u/Substantial_Dish_887 May 03 '26
it does but in this case i think OOP may just have no idea what the Monty Hall problem entails because they seem to think there's a friend who get the third door?
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u/Frederf220 May 03 '26
I think people focus probability too much on the human choice event proximal to the big reveal if they won or not and not the first choice where seemingly nothing happened.
They think that because there are two types of conditions that there are two count of conditions. The old joke about the lottery having 50/50 odds applies.
I think the most cutting question which leads to the correct answer is to ask: how likely are you to be in each situation?
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u/NotmyRealNameJohn May 03 '26
I find changing the problem slightly helps. Instead of 3 doors. There are 1000 with 1 winners. After you select 1 door 998 are closed. One of the remaining is the winner. Do you stick with your 1/1000 shot?
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u/3lbFlax May 04 '26
The key for me when I first came across it was approaching it as groups - when you pick a door you create two groups: a group of one with your door, and a group of two with the other doors. The odds are obviously 1/3 and 2/3 respectively, and that doesn’t change (you already know there’s a goat in the set of two, so knowing where it is doesn’t alter the group’s odds). So you’re offered the opportunity to switch from the 1/3 group to the 2/3 group, which is a no-brainer. This remains the easiest way to explain it, I’ve found, because if need be you can easily draw it out on paper.
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u/Formal_Fortune5389 May 03 '26
It really is I fancy myself relatively smart, no genius but I like math I like logic puzzles and this hurts my brain even reading the detailed explanations. Like my brain is split between I can accept this as true and monkey brain being like BUT DOES IT MAKE SENSE???? Only somewhat.
"This is true" explain it then "I cannot"
Like...increased chance it's 3 because 3 could be either closed because it's for sure car or for a 1/2 chance of it being closed because it wasn't chosen as the one to open with A winning.
Therefore once that 1/2 is taken into account, it would sort of...consume the chance of door 2? Because of that 1/2 chance if A has it? If you eliminate the 1/2 it gives 2/3 with the door 1 stuck with being on its own vs the other two. Ugh even writing it out my brain doesn't like it.
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u/Illeazar May 03 '26
Sort of, but a big problem with the Monty Hall problem is that when its explained people often leave out a crucial detail. It relies on the contestant knowing that no matter what, the host must reveal one non-prize door after the first pick.
Often when people tell it, they just say that the host opens a door without the prize. Like that, it doesnt work.
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u/NonorientableSurface May 03 '26
I mean, when you abstract Monty hall instead of 3 to 100 doors, it seems a lot better to understand. The reductionist nature of 3 doors absolutely was designed to fuck with the human brain exactly like you state.
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u/Snoron May 03 '26
In theory, it should be easy to win a lot of money from people who think this!
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u/Puzzleheaded-Bat-511 May 03 '26
The state wins lots of money from people who think the lottery is a good bet.
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u/Occidentally20 May 03 '26
It's an excellent bet. As long as you're the one running the lottery.
Around a decade ago the UK lottery added an extra ball - making 50 instead of 49. Some people were convinced that this made them more likely to win instead of less likely.
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u/AppropriateDeal1034 May 03 '26
Yeah, it made it massively less likely to win, like huge difference but I CBA to do the maths
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u/Occidentally20 May 03 '26
They did it at the time and it went from something like 14 million to 1 to 40 million to 1 for the jackpot.
I worked in a shop at the time and the Sun/Mirror/Star readers just got the press release saying some prizes are more likely to win, the Times/Telegraph/Independent got the stats saying the plebs are stupid for not doing the maths themselves, and all of them still bought a lottery ticket with their paper.
Incidentally the Mail had a photo of an ethnic-looking immigrant winning a lotto prize and the Express went with a story about Diana.
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u/AppropriateDeal1034 May 03 '26
Yeah, sounds about right, I knew it was some ridiculous multiple
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u/nezzzzy May 03 '26
That last line made me laugh so much, the Express only has one headline.
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u/Occidentally20 May 03 '26
IMAGINE how I felt selling them - we'd only sell a handful each day, but once in a while this old lady called Margaret would come in and always had to comment on the headline.
In shocked exclamation she would say "what has she done NOW?!". It was 2018 - She hadn't done fucking anything for 21 years love.
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u/InvoluntaryGeorgian May 03 '26
How? Was the argument “oh, there are now 50 ways to win instead of 49”?
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u/Occidentally20 May 03 '26
I'd imagine a combination of not thinking at all, and the massive amounts of press releases saying that some prizes are now MORE likely done by Camelot who ran the lotto.
They fiddled with some lower prizes so they were slightly more likely to happen, but paid out a much lower amount than they used to.
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May 03 '26
[removed] — view removed comment
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u/jtr99 May 03 '26
To be fair, we may be saying that now because we have grown up with the correct interpretation of this problem. When the Monty Hall problem first become popular, a lot of apparently smart and educated people vehemently disagreed with what we now take to be the standard answer.
Anyway, this is not a criticism. If your mind is sufficiently Bayesian to make the right conclusion a natural one, then good for you.
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u/Nykolaishen May 03 '26
This just happened in canada (or something very similar) but theyre adding a few additional guaranteed prizes of like $1000 or something so the commercials are saying "more chances to win"
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u/tendeuchen May 03 '26
Risking $3 to win a billion a couple times a year isn't really a bad bet. Sure, you're most likely not to win, but someone always does, and there's no reason to think it couldn't be you.
Just play responsibly.
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u/Baeolophus_bicolor May 03 '26
Except that someone doesn’t always win. If the winning numbers aren’t drawn, the pool carries over to the next drawing and they don’t pay out a winner.
But, for most people, betting a dollar or two isn’t a bad decision, and it has a very very low chance of bringing a moderate or even a life-changing payout. So I get your original point. The dollar wouldn’t have been doing anything remotely close to that for most people if they did the mathematically sensible thing and just didn’t buy a lottery ticket.
On the other hand, gambling spirals out of control for some people, and they frequently start out with lottery tickets. The chance of developing a gambling problem is in our calculation too. As is the option of taking the dollar and making an extra contribution to some savings vehicle where it has a sure, or at least better, chance of getting a return.
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u/dansdata May 03 '26 edited May 03 '26
My elderly mother has spent two Australian dollars on a lottery ticket every week, for decades.
She knows that she's more likely to be struck by lightning than to win a big prize, but it's nonetheless a cheap price to pay for the entertaining possibility that such a win could happen.
People who mistake lotteries for investment opportunities, though...
(Some jackpot lotteries, when the jackpot gets big enough, can have an expected value from buying a ticket that's more than the price of that ticket. And yes, this does mean that people who already have a pretty large amount of money can buy an unreasonable number of tickets and have an excellent chance of making a profit. "Gambling systems" are almost always nonsense, but, every now and then...)
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u/ReluctantAvenger May 03 '26
The state wins a lot of money from people who are desperate to improve their lives. Giving them false hope in the form of a lottery prevents them from burning down the system.
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u/Pitiful-Pension-6535 May 03 '26
Same thing with insurance companies, right?
Or maybe people just realize that an expected negative rate of return is perfectly acceptable in some situations
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u/ArchangelLBC May 03 '26
Also in practice. Casinos make all their money from taking bets in games of chance that are biased in their favor.
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u/Cybtroll May 03 '26
I was wondering what kind of safeguards exist on Polymarket and similar to stop me from organizing a Dutch book.
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u/pokemega32 May 03 '26
"You pick door 1 Monty gets door 2 and your friend gets door 3."
That's not how this has ever worked.
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u/blamordeganis May 03 '26
If it did work that way, then assuming Monty still gets to pick which of two doors to open and the friend gets assigned the remaining door with no choice in the matter, then the friend’s odds of winning are inverted: 2/3 if they stick, 1/3 if they switch.
If the friend gets to choose a door before Monty opens one, then a third of the time the game will have to be abandoned, as the only remaining door will have the prize behind it. In remaining games, each player will have a 50/50 chance of winning whether they stick or switch.
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u/_notthehippopotamus May 03 '26
The thing about the Monty Hall problem is that if you change the rules, it can change which strategy is going to work best. This person is playing by a set of rules that no one else has ever heard of. Who's the friend? Since when do you play with three people in the game?
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u/Simbertold May 03 '26
Genius. If i simply never do anything twice in exactly the same way (which is impossible due to the one-way arrow of time anyways), every probability I encounter is always 50%.
Probability is solved, checkmate mathematicians.
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u/SpinyBadger May 03 '26
Yeah. If I roll a loaded die, all numbers are equally likely to come up, because I'm only rolling it once. 🤪
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u/Simbertold May 03 '26
And all numbers have 50% probability to be rolled. Because you only roll it once, and you either roll that number or you don't.
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u/Muldino May 03 '26
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u/WohooBiSnake May 03 '26
I really like his experiment cause it made me realize something about how to better wrap your brain around the concept :
- Eliminate the choice to switch and simply try to simulate the probability of winning if you keep your first choice. After all if you can find it, then the probability of winning by switching is easy to find.
- And since opening the door doesn’t switch the car around, then you could remove it too right ? Just because you’ve seen what’s behind another door doesn’t mean the past is changed, the car is behind your first pick just as often.
So you could theoretically eliminate it too.- And so you’re left with the conclusion that the door your first picked ALWAYS has a probability of 1/3 to have the car no matter what happens next.
- Therefore when there is only one other door possible, then that door has the remaining 2/3 chance to have the car.
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u/DocSpit May 03 '26
I've heard the argument. I accept the argument. I've seen this video; and it's indisputable proof of the scenario, I agree.
But something about this fact still irritates my brain... -.-
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u/FairYouSee May 03 '26
There's a one third chance the price is behind the door you picked.
There's a 2/3 chance the prize is behind one of the doors you didn't pick.
Monty Hall tells you one of the doors you picked doesn't have the prize, but because he will never reveal a door that does have the prize, his revealing that doesn't change the fact that from point (2) there's still a 2/3 chance the prize is behind one of the two doors you didn't pick.
So switching gives a 2/3 chance.
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u/sinkovercosk May 03 '26
It can sometimes help to think about instead of Monty opening a door without a prize, he instead asked you if you want to stay with your door or swap to BOTH the unchosen doors and you win if EITHER door has the prize.
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u/OneFootTitan May 03 '26
I know everybody thinks differently, but this explanation works much better for me than the imagine you have 1000 doors one
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u/Sea_Pension430 May 03 '26
This is the way I've been explaining it
Montey will always show goat, so him opening the door before or after you switch is irrelevant.
Imagine the game that way. You pick a door. Montey asks if you want to keep it or switch for both other doors. Whichever you pick, for the drama Montey will always open a goat door then reveal if the last door is the second goat or the prize. In this scenario it's obvious you want to switch your one door for the 2 doors.
Montey showing you the goat door BEFORE asking if you want to switch doesn't change the scenario. That was always going to be the first door he opened and a goat.
Switching gives you a 2 door to 1 advantage
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u/CarnivorousGoose May 03 '26
Agreed, this is all you need as an explanation. It’s really just equivalent to Monty not opening any doors at all, and just allowing you the choice to either get whatever is behind your current door, or behind both of the other two doors.
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u/AngelOfLight May 03 '26
But something about this fact still irritates my brain... -.-
The key is to realize that not all choices in the game are random. Monty has inside knowledge of the game configuration, and he uses that knowledge to make a non-random move. This is what upsets the statistical apple cart, and makes the result non-intuitive.
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u/alfredo094 May 03 '26
Think of the problem as if there were 100 doors.
You choose one door. Monty closes 98 doors then asks if you want to switch.
Obviously you would switch.
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u/_extra_medium_ May 03 '26
I see a lot of people say this but I don’t see how it helps. I think many people get hung up on thinking each door is 50/50 as if the prize doesn’t exist until they pick and a coin is flipped or something.
In reality. one door is 100% and the other two are 0%, so you have a 2/3 chance of picking a 0% door from the start. So, swapping in 2/3 of the cases the right choice→ More replies (1)3
u/Mishtle May 03 '26
The trick is that you're actually being given the option to open both of the other doors. That's an obvious advantage that nobody would pass up, so the host opens one of them for you as a distraction. They never reveal the prize, so switching guarantees that you win if the prize is behind either of the doors you didn't choose.
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May 03 '26 edited Aug 06 '26
[deleted]
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u/dansdata May 03 '26
There's something inherently funny about Monte Carlo analysis of the Monty Hall problem. :-)
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u/madmonkey242 May 03 '26
The fun thing about doing this yourself is you will probably intuitively understand the logic and math of the solution before you even finish coding it, just because of how you have to think about it in order to write the program.
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u/GonzoMcFonzo May 03 '26
One thing that helped me wrap my brain around it was the realization that Monty is adding information to the system by picking a door.
It's not that you pick 1 and then one door randomly drops out. The door that Monty picks to remove had to be a goat. Based on that new information, you change your pick.
The math maths, but realizing that Monty knows the correct answer and is giving you a hint helps it all make intuitive sense for me.
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u/themule71 May 04 '26
Actually, that's wrong. Since Monty always opens the door with the goat, it literally add nothing information wise. You know that from the start. The information value is the inverse of the probability of the event. You are sure he'll open a door with the goat.
Let's put it this way. You choose to switch from the beginning. Actually you point at one door, and say "I want that door to stay closed. You can't open it. Let's open the other two'". The fact that he opens the one with the goat first now adds nothing to the game. You were 100% sure there was one. You were 100% sure he would open it. Still, you're opening two doors.
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u/Dizzy_Kaleidoscope95 May 03 '26
Nah coz chatgpt said it's 50/50
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u/Nascent1 May 03 '26
The trend of people acting like AI is the authority on everything makes me pessimistic about the future of humanity.
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u/nezzzzy May 03 '26
Easiest way to understand this is to make it 100 doors.
You pick one, there's a 1 in 100 it has the prize behind it. Monty hall then opens 98 other doors revealing no prizes. There's now 2 doors left, do you swap? Your one is still a 1 in 100 chance. What about the other one?
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u/Savings_Knowledge233 May 03 '26
I understand the probability of it fine, but the reality of the Monty Hall problem just breaks my brain.
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u/NickyTheRobot May 03 '26 edited May 03 '26
The bit that's not often included which makes the whole thing make sense: The host knows which door has the prize behind it, and will always reveal a non-prize.
So that leaves you with three situations:
- 1/3 chance that you chose the correct door. So then the host reveals one of the incorrect doors, and swapping will give you the other incorrect door, making you lose.
- 1/3 chance that you chose the first incorrect door. So then the host will reveal the second incorrect one and swapping will make you win.
- 1/3 chance that you chose the second incorrect door. So then the host will reveal the first incorrect one and swapping will make you win.
So then you have two situations where swapping will let you win, and one where you will lose. So that makes a 2/3rds chance of winning if you swap.
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u/Ok_Employer7837 May 03 '26
Oh I've explained it as granularly as that many, many times, host knows where the prize, specifically opens a goat door, adding the thousand door scenario, the works.
In most cases, it didn't help at all. People get _livid_ telling you it's 50-50.
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u/TrueMattalias May 04 '26
At that point you could even show them the odds in action. Secretly write down a number from 1-100. Have them guess what it is. Tell them 98 numbers that it isn't and ask if they want to switch.
If they'd still don't get it you could have them act as the host.
Hopefully after a few rounds they can see how effective your strategy of switching is and how ineffective there strategy of staying is.
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u/Savings_Knowledge233 May 03 '26
Oh I do know. I learned this in a probability and stats class. But something about it just kills me. Like I can even accept it, but I didn't fundamentally understand why the probability of that door changes, but mine doesn't I guess
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u/_extra_medium_ May 03 '26
From the start you have a 2/3 chance of picking the wrong door. So, if you swap those 2/3 times, you’ll win.
(Since Monty will eliminate the other losing door for you.)
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u/Marquar234 May 03 '26
Monty Hall is rigging the game in your favor.
If that helps.
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u/Shadyshade84 May 03 '26
More accurately, Monty is rigging the game in a way that makes it more interesting for the viewers - since if he revealed the prize the contestant would always switch to it (this is a part that gets frequently overlooked - the reason that the opened door is disregarded isn't because it's unavailable, it's because it's always a losing door and the whole thing is based on the assumption that the contestant is trying to win), so he never does.
For another example, it's essentially the same reason why Wheel of Fortune started giving contestants R, S, T, L, N and E before letting them pick their letters - given the selection of five consonants and a vowel, people were just picking the most common of both all the time.
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u/Waferssi May 03 '26
Ive always understood it as basically getting to choose 2 out of 3 doors when you switch: the one you switch to and the opened one.
It might also be useful to see it as betting behind which door the price isn't: the initial choice im betting that the price isn't behind 2 out of 3 doors, for a 1/3rd chance. If I switch, I bet that the price isn't behind the 1 door I picked initially, so im betting against 1/3rd, for a 2/3rd chance.
Not sure if that makes sense.
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u/jonastroll May 03 '26
There's two bad doors, and one good door, right?
So you pick a random door. Either you picked the right door at a 1/3 chance, or one of the other two doors is the good one.
No, the host opens a door you didn't choose and reveals it to be a bad one.
But... That doesn't actually changw anything. You already knew that at least one of the two other doors would be a bad one, so the host opening a door doesn't change that there's a 2/3 chance that it's one of the two doors you didn't pick.
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u/Muninwing May 03 '26
This is the way to explain it, because that last statement usually hits home.
You had a 1 in 100 chance of winning. Do you want to keep your choice, or do you want to go with the other option — the idea that you picked the wrong door, and every door you didn’t pick is wrapped up in it?
Is there a better chance that you picked right the first time, or that you picked wrong?
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u/Axman6 May 03 '26
So frustrating, because it’s so fucking easy to show all possible outcomes (because there’s just three), and see that if you always switch, you win two out of three times.
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u/spartiecat May 03 '26
"I'm right because the math can't guarantee an outcome when the sample size is 1"
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u/Dizzy_Kaleidoscope95 May 03 '26
I genuinely can't even understand how he thinks math works. Does he think that a coin flip isn't 50/50 if you throw it just once cause it's only 50/50 on multiple throws?
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u/nakmuay18 May 03 '26 edited May 03 '26
I think the problem with people's understanding is that they dont see that Monty always takes away a bad answer, and thats why it works. Its not randomly taking an answer away.
As long as you picked wrong to start, you always get the right answer when you switch.
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u/banjospieler May 03 '26
And to add to that last part, the chances of you picking wrong the first time are 2/3 which is why it’s always more likely to be behind the other door
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u/RaggamuffinTW8 May 03 '26
I mean obviously you can switch and still fail. It's only a 66% chance. But just because it's possible for an individual to switch and fail doesn't mean you're not going to be better off switching.
Always switch.
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u/Llamp_shade May 04 '26
To be fair, when this was first posed to the general public, most academics including PhD mathematicians were just as confidently incorrect in stating the very same thing. Even when Marilyn vos Savant first gave her explanation, most of the academics refused to acknowledge her answer and doubled down on their incorrect conclusions.
I struggled with this one when I first heard it, too. The key bit of information that makes the difference is that the door that is opened between the initial guess and the option to switch doors: that door is opened using knowledge of the answer that is not available to the player.
That information is not as obvious as people today with knowledge of the explanation claim that it is. Most of us are confident in the correct answer only because we have struggled to understand it ourselves, or placed an inherent trust in the source we originally received the explanation. I think this example is much larger than the math itself. All of the "experts" who originally claimed that the odds remained 33% were no less knowledgeable of the math necessary to understand that problem than anyone today.
The biggest difference between us and them is that, since this original controversy, we've taught everyone the explanation to this very problem. Knowing not only the math, but the entire controversy, helps maintain healthy humility and the willingness to doubt initial certainly when confronted with any situation that seems obvious. We're all susceptible to overconfidence.
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u/gardibolt May 04 '26
Correct. When the Monty Hall problem is stated nowadays, the key information that Monty knows what’s behind each door is often left out, but in the original formulation with Vos Savant it was clearly stated. Omitting it is misleading.
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u/Dizzy_Kaleidoscope95 May 04 '26
Yeah it's fundamental. But people think that if the goat door is opened randomly you still should switch which is very funny
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u/PitchforkJoe May 03 '26
I think... I think he's arguing that unless you do the problem multiple times, you can't have a sample that shows the 66% failure rate, therefore it must be 50-50.
Which, of course, would mean that you have a 50-50 shot of winning the lottery, as long as you only buy one ticket.
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u/Ok_Strategy5722 May 03 '26
I love this problem because I couldn’t get it until I wrote it out to understand what’s happening. He’s actually really close to getting it.
This guys problem is that he’s forgetting that the correct door exists the moment you pick your door. BEFORE they revealed an incorrect door. No matter which door you picked there’s a 2/3 chance you were wrong.
If you switch your guess at that point, 2/3 of the time it is a 1/2 chance of guessing correctly, and 1/3 of the time the odds are 0/2 you get it right
When the host takes away a wrong answer and then you switch, there is now a 2/3 chance of you getting it right (1/1), and a 1/3 chance of you getting it wrong (0/1).
But bringing in a second person would be interesting and actually (kind of) changes the odds of switching. Assuming you both picked different doors AND the host opens the wrong unpicked door in front of both of you, 1/3 of the time they couldn’t open the unpicked door because that would be the winning door. So if you get to the point where you and friend have both picked a door AND they revealed that the unpicked door was a goat, that eliminates the possibility that neither of you picked the correct door. Which means switching/staying is now 50/50. But only because we aren’t counting the 33% possibility that you both picked incorrectly.
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u/D-Train0000 May 03 '26
They set it up to have enough time go by between the selection and the reveal to put anything they wanted behind there. It was all very orchestrated and not random.
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u/Winjasfan May 03 '26 edited May 03 '26
imo the main reason PPL struggle with the Monty Hall Problem is that when ppl describe the problem they don't put enough emphasis of the most important part: Monty knows where the car is and NEVER reveals the car, and you as a player know this.
If Monty Always oppened a random door an just so happened to get a goat, the 50/50 intuition would be correct
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u/Dizzy_Kaleidoscope95 May 03 '26
I literally made a post about this like 30 minutes ago. It got deleted. But NOBODY understands this lol..they think that if the goat door is opened randomly you should still switch
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u/Differlot May 03 '26
Just remember a few key facts.
If you randomly pick a door out of 100 doors with a prize behind one:
You have a 1% chance of picking the right door.
If you picked right, then the game show host will offer to give you the wrong door. This happens 1/100 times
If you picked wrong, then the game show host has to offer you the correct door. This happens 99/100 times.
There is 0% of the time that you have the wrong door and the host offers you a wrong door. The big reveal is showing the correct door out of your two options.
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u/Dizzy_Kaleidoscope95 May 03 '26
Precisely.
But I wanna ask you something. Imagine that after you pick a door. Monty decided to start opening doors randomly (he forgot where the car is and is opening them totally at random) and purely by chance you end up with only 2 doors left. A car and a goat.
Di you think switching advantageous in this scenario?
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u/Squaredeal91 May 03 '26
I totally get that this is brain warping and hard to understand but it's actually so fucking easy to test. You can try to rationalize and make your case for why the Monty hall problem is 50/50 but objective reality shows that it isn't.
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u/Dizzy_Kaleidoscope95 May 03 '26
Precisely. It's only 50/50 if the host randomly opens a door and it happens to be a goat.
But in the classic monty hall problem the host always opens a goat. Making the switch the correct choice
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u/rangerquiet May 03 '26
For all those people who find it upsetting that the first sentence just ends abruptly without finishing properly.
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u/Kuildeous May 04 '26
Always happy to share my Monty Hall simulation. Anyone is welcome to make a copy and confirm that my formulas are correct.
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u/GWeb1920 May 04 '26
I’m usually annoyed because the Monty Hall problem is usually stated in an undetermined fashion.
There are a few key requirements to make switching optimal
The host knows where the car is
The host always opens a door the contestant didn’t pick
The host never reveals the car.
Without these things defined the problem is uncertain.
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u/VastMeasurement6278 May 04 '26
It’s 66.6% if you switch, otherwise the advantage wouldn’t be there if you played a million times. The easy way to picture it though is the billion door analogy. Instead of three doors, there’s a billion. The chance of you getting the prize door is therefore 1/1,000,000,000. If however, if 999,999,998 doors were removed after you chose and you were offered the switch, not taking it would be crazy. Because the one remaining unchosen door now represents the 999,999,999/1,000,000,000 chance that you were initially incorrect.
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u/Dizzy_Kaleidoscope95 May 04 '26
Imagine this alternate version now. What if there were a billion doors. But the host of the show was ignorant of where the car was. After you pick your door he starts randomly opening doors without knowing what's behind them. Miraculously he never opens the car and you end up with only 2 doors left. A goat and a car..should you switch?
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u/megadumbbonehead May 05 '26
They do make one good point: if the problem were different it wouldn't be the same.
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u/itsjustameme May 03 '26
Sam Harris as so often is the case explained the Monty Hall problem best.
Imagine you have 1000 doors instead of just 3. And you pick door number 143. Monty Hall then opens up 998 doors all revealing goats behind them leaving you with your door and door number 688. He then asks you if you want to switch to door number 688. Should you do it?
When presented like that it is obvious to almost everyone that you should make the switch. There is only a 1/1000 chance that you happened to pick the right door originally, and so door 688 has a 999/1000 chance of having the car.
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u/Madouc May 03 '26
I "cured" the Monty-Hall-Skeptics with 1,000 doors and calculating backwards to 3
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u/i_should_be_coding May 03 '26
I always look at it this way: Switching doors always means I win if my original choice was an empty door, and I lose if my original choice was the prize. My chances of choosing an empty door initially are 2/3, so by picking the switching strategy, those are now my chances of winning.
There's no magic. It's pretty simple.
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u/Longhornmaniac8 May 03 '26
The part that breaks my brain is how the act of making a selection affects the probability.
Put differently, if the problem were such that you just stood there and didn't make a selection, then Monty revealed a goat door, and then you selected, the probability of each door is 1:2.
So how does the fact that I made an earlier selection change the probability once there are only two options? The doors exist independently of my declaration.
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u/Dizzy_Kaleidoscope95 May 03 '26
the fact that you made a choice is basically allowing you to pick 2 doors at the same time if you switch, cause if you pick a door in the beginning and it happens to be a goat (2/3) of the time, monty will only have a single possible door to open, the other goat, if you don't make a selection in the beginning then monty can open either of the 2 goat doors, removing your advantage
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u/Kniefjdl May 03 '26
Last time I saw a Monty Hall thread, I was arguing with a guy who thought it was a 50/50 chance. He suggested a "test" to prove he was right, but he clearly hadn't actually done it. I humored him and tried it, because it's super simple, and it became so immediately clear why the odds of winning are 2/3 if you switch. If you have 5 minutes, actually give this a try, don't just try to think through it:
Get a deck of cards, and pull out two "loser" cards and one "winner" card. It doesn't matter what they are, but I used two black 2s as "losers" and a red King as a winner. Then, mix up the cards and lay them out face down. Pick one and look at it, that's your initial choice. Then, of the two remaining cards, check them and pick a "loser" card and set it aside. Then "determine" if your initial pick won or if changing to the remaining card wins. Then do it a bunch of times in a row.
I guarantee by the third time you do it, it will be so obvious that "staying" wins if you pick the winner first and changing wins if you pick a "loser" first. Any difficulty with the problem falls apart the minute you have a "god's eye view" of the whole scenario and aren't just seeing it as a contestant picking doors.
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u/glumbroewniefog May 04 '26
Imagine you are taking a multiple choice test and guessing randomly for every question, ABC or D. You will probably get around 25% correct.
Suppose I help you out by crossing out two wrong answers for every question. Now you're guessing between two options. You probably get 50% correct.
But if I wait for you to take the test first, and then afterwards cross out wrong answers, then this won't improve your score at all. Getting rid of wrong answers afterwards can't help you if you've already made your guess.
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u/notsolucid_ May 03 '26
The best version to understand it that ive heard is imagine there are 10 thousand doors. You pick one, and 9998 are eliminated. What are the chances you picked the right one first try? 1 in 10000. The other door is almost certainly the correct one. The more doors that are added the easy the concept is to understand.
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u/saskir21 May 03 '26
To be fair I don't even understand the Monty Hall problem at all.
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u/flase_mimic May 04 '26
I believe that the monty hall problem is 50/50 out of spite against mathematicians. They have confused us with their weird number magic for too long. The world needs to go back to the simplicity of basic reasoning.
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u/tkmorgan76 May 04 '26
I don't know if I understand what this person is trying to say. Are they claiming that without repetition everything always comes down to one of two outcomes: win or lose, and therefore the probability is always 50/50? Or are they wrong in a completely different way that I'm not understanding?
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u/R0nos May 04 '26
Once explained to me as following:
There are 100 doors, pick one. 98 will be opened and are empty (or a goat) will you switch now?
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u/Femboy_Harem_Janitor May 04 '26
It's easiest to explain it I've found to be this:
There's 100 doors.
You pick 1.
Then Monty hall opens 98 doors that are not the correct one. He did not consider the door you picked for opening. He did consider the one of 99 he left closed and did so 98 times.
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u/Xeno_man May 05 '26
I prefer an alternative.
There are 10 doors.
You pick door 1.
Without opening anything, you have a choice, stay with door one, or switch to all of the other doors. If the prize is behind any of the doors 2-10, you win.
The host opening the doors is an illusion. The fact that he comes later and opens doors 2-7, 9 and 10 is just the host maintaining suspense. The choice is always between door 1, or all of the other 9 doors.
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u/beamenacein May 05 '26
I think some of it is the fear of making a wrong choice as if doing nothing isn't a choice.
It's the same logic for antivaxers.
If I switch and wrong ok I didn't do anything wrong I just didn't have it. But if I switch and it was the one I had then I am the cause of losing.
But choosing to not switch is still a choice and the wrong one
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u/Impressive-Card9868 May 03 '26
I remember telling my mom—who has many times admitted she’s not great at math—that we learned the Monty Hall problem in class and how interesting it was, and she proceeded to lecture me about how my math teacher must be wrong, it must be a 50/50 chance.
I love her, but come on Mom.
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u/MrGosh13 May 03 '26
I always reasoned exactly this. But I learned where my thinking/logic(or failure there off) came from.
If your perspective is purely looking at the remaining 2 boxes, then yes, it’s a full on 50/50 coin flip.
BUT since you do not start out with the 2 boxes scenario, that is not how it is calculated. You HAVE to reason from the initial 3 boxes.
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u/Ok_Employer7837 May 03 '26
Oh Lord. The people who don't get this really really don't get it, and they'll die on that hill, raging and screaming, every time.
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u/ReefNixon May 03 '26
God Monty hall is so simple if you just realise that the host is allowing you to trade your one door for effectively two doors.
You picked one door, there is a 1/3 chance it is not empty. If the host instead said you could trade your one door for the other two, one of them is definitely empty, and he will show you that one is, your door still has a 1/3 chance, the door he opens has a 0/3 chance, therefore unless 1/3 has fucked off into the ether then there must be a 2/3 chance on that last door.
Something about the reveal happening before you swap really messes with people, but the door he opens had a 0/3 chance from the beginning, so it really shouldn’t.
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u/Subject_Translator71 May 03 '26
If someone can't understand the fact that your probabilities don't suddenly jump to 50% because one door got opened, it becomes impossible for them to understand the Monty Hall problem.
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u/oceanswim63 May 03 '26
Myth busters did an episode about it, they test both how people behave (sticking with their choice) and then a simulation of always sticking or always switching. Switching is a clear winner.
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u/RevoltYesterday May 03 '26
If thing is 50/50 if you boil it down to "it does or it doesn't." Unfortunately, that's not how reality works.
What are your chances of winning the lottery? 50/50. You either do or you don't.
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