The Monty Hall Problem, a fun little game

hasher22 on 03/02/2022 - 20:03
Last edited 03/02/2022 - 20:05

This is not a riddle, but about probability. Analysing the words, door selection, and structure of what I said will not help you at all

Let's play!

You are at a game show, there are 3 doors in front of you but you don't know what's behind them. Behind 2 doors are goats, behind 1 door is a car. Your goal is to select the door with the car and you win the car! You have a 33% chance to win the car and a 66% chance to select a door with a goat.

You select door 3
The game show host reveals door 2 which has a goat
The host now asks you, "Do you want to switch to door 1 or stay at your current door, 3?"

Question: What do you do? Do you switch doors or stay at your selected door to possibly win the car?

Comment below why.

Poll Options

59
Stay at door 3
321
Switch to door 1

General Discussion

Comments

DiscoJango on 06/02/2022 - 18:03

-1

Isnt it 50/50 either way and switching or staying makes no difference?
- CrowReally on 06/02/2022 - 22:21
  
  +1
  
  Yes, that's what it says in all the responses to this thread.
- CanadaAye on 06/02/2022 - 22:34
  
  +1
  
  No, switching gives you a 2/3 chance instead of the 1/3 you had with your initial choice.
  
  It may be easier to think of it this way; switching is effectively allowing you to choose both of the other options that you didn't choose to begin with. At least one is always going to be a goat, and the host will reveal this to you.
- filmer on 07/02/2022 - 09:15
  
  +1
  
  If the host chose a random door its 50/50. If a host chose a goat door on purpose it's 66% if you swap.
NuclearWessels on 06/02/2022 - 18:07

+3

A simple way of looking at it- if the host is always removing a dud prize, then you should always switch. Switching in this way inverts your answer - if you'd picked a losing door you now win and if you picked a winning door you now lose. The initial probability to lose was 2/3 - so inverting it by switching becomes a 2/3 win chance.

If you do a truth table it becomes very clear why switching works.
AnophthalmiaCervidae on 07/02/2022 - 10:53

For anyone still struggling.
If you were offered to switch your original door with the other two doors (before the goat is revealed) you would always do it.
Opening one of your two swap doors (that the host knows does not contain a car) does not change the original odds, you still have a 2/3 chance of winning the car, 1/3 chance it was behind the original door.
Same odds as if the goat is revealed before you switch.
The host opening a door with a goat is not a random event, there is a goat behind at least one of the two doors and the host knows which one to pick.
- nobro25 on 07/02/2022 - 18:32
  
  The fact that you're saying 2/3 is winning the car is false. Because if that was the case then there'll be two cars to begin with. Making the odds in your favour. It's always 1/3 chances because all three doors are unknown.
  
  Then it becomes 1/2 chances simply because there's really two options left after the third is revealed. If you keep saying 1/3 or 2/3 after the third door is revealed then you're implying that even after revealing the goat door, you can still pick the door that is revealed. Which is stupid.
  
  You actually only have two choices which makes it 50/50. Think about a coin flip, each flip is independent of the previous flip. Just because you scored 10 heads in a row the next one is not necessarily a tail.
  - AnophthalmiaCervidae on 07/02/2022 - 20:03
    
    +2
    
    Still not quite getting it are you?
    
    If you pick one door you have 1/3 chance of picking the car. If you had a pick of two doors you would have 2/3 chance of having picked the car, and a 1/3 chance that you don't. That does not change if a non winning door is revealed (non random).
    
    If the door reveal is completely random then yes it is then a 50/50 after the reveal but 1/3 of the time the deal would fall over at that point as it would have revealed the car.
    
    Your chance of winning is still 2/3 x 0.5 = 1/3 because 1/3 of the time you would just be swapping one goat for the other.
    
    By swapping after the host has revealed a door they know to have a goat you effectively get to open 2 doors. There was a 2/3 chance that one of the two doors you didn't pick had the car, it does not change the odds after the non car is revealed.
Keeegs on 07/02/2022 - 11:38

+1

Jokes on you, I want a goat.

Donaldhump on 07/02/2022 - 16:45

python code proves chances are double.

10000 simulations

0.3377

0.6666

import random
C_NUM_GAMES = 10000 # num games
C_DEBUG=0
C_NUM_DOORS=3
m = 1 # two modes, 1 for not switch, 2 for switch

while m<=2:
c=1
won = 0
while c<=C_NUM_GAMES:
l = ['G'] * C_NUM_DOORS
r = random.randint(0,C_NUM_DOORS-1)
l[r] = 'C'
res='L'
player_pick = random.randint(0,C_NUM_DOORS-1)
#print(player_pick)

    host_pick = player_pick
    while host_pick == player_pick or l[host_pick] != 'G':
       host_pick = random.randint(0,C_NUM_DOORS-1)

    res ='L'
    player_pick_save = -1 
    if m == 2: # switch
       player_pick_save = player_pick
       while player_pick_save==player_pick or player_pick==host_pick:
           player_pick = random.randint(0,C_NUM_DOORS-1)

    if l[player_pick] == 'C':
            res='W'
            won=won+1     
    if C_DEBUG==1:
        print('========================')
        print('mode = '+str(m))
        print('host pick='+str(host_pick))
        if m == 2: # switch
            print('player_pick_save ='+str(player_pick_save))  
        print('player_pick ='+str(player_pick))  
        print(res)
        print('========================')
    c=c+1
m=m+1    
print(won / C_NUM_GAMES)

hasher22 on 07/02/2022 - 18:21

That's deep

Michael074 on 07/02/2022 - 18:11

+1

I did computer science and second year probability at uni and im ashamed to say it took me a good hour to fully understand this problem. its easy to manually test to find the answer but this does not mean you completely understand. based on personal experience I'd say there are 2 reasons why it is difficult to understand the solution. 1. its worded and framed in a way that is totally non intuitive. 2. most people are terrible at explaining it probably myself included. the reason i understood is because I eventually found an explanation on the internet that made a lot of sense. I spent probably 15-20 minutes at a time with people trying to explain it or reading other explanations and made no progress, but once i had a good explanation it immediately clicked. this was a while ago but ill do my best to give the most simple explanation possible that at least for me was the key to getting the solution:

after your initial choice you are in one of two different worlds

A) you chose the car, one goat is eliminated, if you switch you get the goat.

B) you chose a goat, one goat is eliminated, if you switch you get the car.

at this point this is where people make their mistake, they think so theres 2 different scenarios so its a 50/50 chance. wrong. why? because you just forgot that 2/3 times you will be in world B. since you don't know which world you are in, its best to assume that you are in world B because you have the greatest chance of being there. and the winning strategy if you are in world B is obviously to switch. so you always switch. only 1/3 times you will be in World A making a switch a mistake.
- hasher22 on 07/02/2022 - 18:19
  
  The thing is, converters or people who understand it later on such as yourself, myself and many others will need to read many examples in different formats to fully understand this.
  
  I, myself, was a 50/50 person, no matter how many examples or explanations I read, I still didn't understand it until I made my own Study using 3 physical cards and labelling one each of G G C. If I pick up a goat card, I switch and I get 1 point under "switch". If I get a car card, I won and 1 point under "first pick". My results, funny enough was 66percent switch, 33 percent first pick. I did it 30 times
  
  That's when I fully got this problem then everything made sense of what people were talking about.
  
  I think with this problem, people have vastly different ways of learning this problem and seeing that it's not a 50/50 chance in round 2.
  
  My example of 1000 doors gives it more clarity on why someone should switch.
  - Michael074 on 07/02/2022 - 18:27
    
    I did that too. but i feel like i wasn't 100% comfortable with the result until I realized that it would be a 50/50 chance, except the door you currently have selected because of your previous choice has a 2/3 chance of being wrong.
- nobro25 on 07/02/2022 - 18:25
  
  Still makes no sense though. By your logic if the same scenario with 4 doors and to however many doors is 1/3 then.
  So you pick one door out of four. Then they reveal two doors as goats. Do you stay or switch? So you'll switch in that situation as well?
  - Michael074 on 07/02/2022 - 18:31
    
    +1
    
    I'll solve your version of the problem with the same logic. after your first choice you are in two worlds:
    
    A) you chose the car, two goats are eliminated, if you switch you get a goat.
    
    B) you chose a goat, two goats are eliminated, if you switch you get a car.
    
    3/4 times you will have chosen wrong and be in world B. but since you can't know which world you are in you have to assume that you have in fact chosen wrong. so you should switch. then 3/4 times you will get the car.
- chuckbass on 07/02/2022 - 21:28
  
  +1
  
  I've spent days trying to wrap my head around why it does not result in a 50/50 chance, and only finally understood after reading your explanation! Thank you!!
Applause on 07/02/2022 - 18:28

I remember this problem from this video https://youtu.be/ggDQXlinbME
haemolysis on 07/02/2022 - 19:03

This is “problem” is only interesting insofar as it is presented as a quirk or mathematics, but it actually just a quirk of premises.

The premises are either that the host picks doors randomly, or the hosts picks them with wilful intent.

Most people who don’t “get it” tend to be thinking about random distribution. For instance you could map out every single scenario and it wouldn’t make any sense to swap doors.

But the map of the probability space won’t factor in that the host has knowingly eliminated from a choice of two doors, where you’ve guessed from a choice of three.
- Autonomic on 08/02/2022 - 16:15
  
  he premises are either that the host picks doors randomly, or the hosts picks them with wilful intent.
  
  The premise is the host reveals a goat. If the host reveals the car, the entire question of switching doors is moot.
allegiance86 on 07/02/2022 - 19:34

Saw this on Mythbusters years ago.

Was completely boggled at the time.
The Wololo Wombat on 07/02/2022 - 21:02

+1

Lol all the people here sharing what the TedEd video tell yous and pretending it's their original thought.
rogerwilko on 07/02/2022 - 22:42

+1

The spoon is bent only if you perceive so.
liongalahad on 08/02/2022 - 03:13

Changing choice increases your chances to win.
For whoever is skeptical here, here's a simulator where you can test it and see the results with your eyes. https://www.mathwarehouse.com/monty-hall-simulation-online/
Taro Milk Tea on 08/02/2022 - 10:06

Should have put the movie reference if you want to Steal this question
- hasher22 on 08/02/2022 - 12:28
  
  the movie 21 was released in 2008, The Monty Hall Problem existed in 1975.
rlf on 08/02/2022 - 11:30

People need to learn about Bayes theorem.
dangerous on 08/02/2022 - 13:24

+1

This question has been asked incorrectly. It only makes sense to switch if it's clear that the gameshow host will only open a door which has a goat behind it.
- Autonomic on 08/02/2022 - 16:16
  
  +1
  
  If the host reveals a car then the entire question of switching is moot.
- Qazxswec on 08/02/2022 - 19:50
  
  +1
  
  The host does only ever open the door with the goat but, why would he ever open the door with the car
- hasher22 on 09/02/2022 - 17:21
  
  That's the point of the game, the host will always reveal a door with a goat the player has not selected, there's no point revealing the player's door or the door with the car in chance the player may have the door with the car. It will always be a non-chosen door with a goat.
sathion on 08/02/2022 - 14:09

https://www.youtube.com/watch?v=iBdjqtR2iK4 Sums up the Goat and Car problem
maximum on 08/02/2022 - 15:52

+1

Where is the poll option for choosing door number 2? What could be better than a guaranteed goat.
Wiired on 08/02/2022 - 19:24

Interesting thing, was watching a few episodes of the original show and they had hidden prizes behind and inside the set of some "goat" prizes so really they could change it up however they liked. They also had two people playing for different doors so it didn't follow any hard and fast rules, that I can see.
Dancasper on 08/02/2022 - 20:04

-1

Have only briefly scrolled through the responses but am boggled by the responses.

Your first pick is either a car or a goat. One of the goats is removed and you get to choose again.

It does not matter what you picked originally, you are left with a choice between 2 doors.

From this point on, you have a 50% chance of winning. The statistical odds do not accrue.
- hasher22 on 08/02/2022 - 20:05
  
  +1
  
  Maybe have another scroll through the comments and you'll find out why 50% is wrong.
  - Dancasper on 08/02/2022 - 21:58
    
    +1
    
    OK. I drew a probability chart to map out all the outcomes.
    
    Your initial choice presents 3 outcomes (1/3 probability it is a car or 2/3 it is a goat, behind the door you picked). Your second choice is stay or switch. All up, this yields 6 possible outcomes.
    
    If you opt stay, only 1 of the 3 possible outcomes will reveal a car.
    If you opt switch, 2 of the 3 possible outcomes will reveal a car.
    
    You therefore have greater probability to win the car by doing the switch.
    
    Wikipedia explained that you have 1/3 chance to pick the car from the initial choice. Whatever door you pick, there is 2/3 chance that the car is behind the remaining doors. The host does you a favour by removing one of those remaining doors (shows the goat)! Therefore the remaining door presents as 2/3 chance of being a car. If you stick with the door you picked, it only has 1/3 chance of being a car.
    
    Can i change my poll choice :)
    - DiscountForThee on 09/02/2022 - 12:34
      
      +1
      
      It's so lovely seeing someone figure it out completely on their own :)
      You can't be blamed not understanding it on first pass. It is tricky to conceptualise, hence why the problem has its own wiki and why it existed on the tv show without people realising.
goedkoop on 09/02/2022 - 02:28

-1

OP has omitted one critical detail of the Monty Hall problem - Monty is committed to always revealing a goat behind one of the other two doors no matter which door you select as the contestant. Since Monty is committed to always revealing goat, it is better to switch (2/3 vs 1/3).

Without this detail, the probability is 1/2 vs 1/2. It would be the same as if, in Deal or No Deal, Andrew let you swap to the other briefcase when there's only two left unopened. The expected value of swapping would be 0.
- hasher22 on 09/02/2022 - 17:24
  
  The host will always reveal the door with the goat which is not the player's selection. There is no point opening the player's door or the door with the car because there is a 33% chance they selected the car, hence this problem wouldn't exist.
  
  For example:
  1. If Monty opened the player's door, there is a 33% chance they won and the game is over.
  2. If Monty opened the player's door, there is a 66% chance it's a goat and if Monty asked to select another door, this problem wouldn't exist cause the point of the game is to switch in the second round, not to re-select.
  
  Monty will never open the player's door selection.
  - goedkoop on 10/02/2022 - 04:32
    
    Yes - If you know the host always opens a goat door no matter what you chose, you improve your chances of winning car by switching.
    OP didn't make it clear that this is something the host always does. E.g. suppose Monty decides to only reveal a goat door when the contestant has initially chosen a car door. Then clearly you never want to switch! The detail matters.
    - hasher22 on 10/02/2022 - 10:27
      
      -2
      
      It's irrelevant information.
      
      The post does not need to include:
      1. if the player chose a goat or car door
      2. If the host always reveals a goat door.
      
      The post clearly implies a goat door will always been opened and a switch will always be the final question to the player.
      
      If you look at all the comments. Youre the only one with this critique but hundreds of other people got it.
      
      If your information was needed in this problem, then it would have been called out a long time ago and making the problem invalid.
      - goedkoop on 11/02/2022 - 02:54
        
        @hasher22: The post clearly implies a goat door will always been opened and a switch will always be the final question to the player.
        
        Read your post again mate. Nowhere does it say that Monty will always open a goat door. It just said he's opened a goat door in this instance.
        
        The post does not need to include:
        2. If the host always reveals a goat door.
        
        But you need that assumption to get the 2/3. E.g. imagine he reveals a (goat) door if and only if you've chosen the car, then switching gives you a 0% chance of winning. Or imagine he always reveals a completely random door, then switching after seeing a goat gives you a 50% chance of winning. The fact he's happened to open a goat door on this particular instance doesn't tell us this is something he always does, that's something you need to tell us. This was one of the reasons the original problem in the newspaper kicked up so much fuss, it left Monty's behavior ambiguous when his behaviour actually changes the answer. You haven't provided this information in your post, and plenty of people have called you out for it in the comments.
        
        EDIT: Straight from the wiki page, the 3 requirements you need to get the 2/3 vs 1/3:
        * The host must always open a door that was not picked by the contestant.
        * The host must always open a door to reveal a goat and never the car.
        * The host must always offer the chance to switch between the originally chosen door and the remaining closed door.
        
        hasher22 on 11/02/2022 - 16:15
        
        -2
        
        @goedkoop: Again, if hundreds of people got my post and you're the only one in this forum that has a problem with the wording cause it doesn't have the word "always" or it's not up to your vocabulary standards.
        
        No one else has commented that it needs to stipulate it needs to have the word "always" as the post implies and suggests a goat door which is not the player's choice will be opened and the question of to "switch" not "select", will be asked.
        
        Also, the 3 steps I wrote, it state the host opened the non-players-selected door.
        
        So, maybe ask yourself, why can hundreds of people understand this post, but you, the only person that has a problem with it not having "always"? In saying that, this sounds like a you problem.
        
        aragornelessar on 11/02/2022 - 20:57
        
        @hasher22: Actually a number of people have said that your post is ambiguous in the assumptions. You've listed the steps in a way that just seems like this is what happens chronologically - 1st you choose, 2nd host opens a door (which doesn't specify why or reasoning), 3rd you get given a choice.
        
        The wiki points out the different results if different assumptions/rules are made - and your post doesn't specifically call out which rule is applicable here.
        https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_…
        
        Most people have probably understood your post to be the standard assumptions - and possibly because they've heard of your titled "Monty Hall Problem"?
        https://en.wikipedia.org/wiki/Monty_Hall_problem#Standard_as…
        
        hasher22 on 11/02/2022 - 22:16
        
        -2
        
        @aragornelessar: Those details aren't needed and are complementary information and by all means, this post is to have the reader (you) thinking you're playing the game not analyse the game and to make an informed decision.
        
        Let's say you're playing the game right now in the studio, you're not going to be told about every detail in the game such as "The host always reveals a goat and always offers a switch". You're going to step up to the host and play through my steps 1 through 3. The only information before step 1 is that there is a car behind 1 door and goats behind 2 doors, and for you to select a door with the car to win, then step 1 will begin.
        
        This post is to try to replicate that, not add additional information to help the reader / player come to a more definitive answer or analyse the problem on a deeper level.
        
        Even if the title isn't "Monty Hall Problem" and it was titled something like "would you switch in this problem? Why?", my post will still state the same question at the end (to switch?) and the same probability will still occur whether or not its the Monty Hall problem or not.
        
        The point was to have readers come to an answer straight away and select an answer. Whether or not they knew the host always opens a door with a goat which is not their door does not change the readers answer cause steps 1 through 3 indirectly suggest that.
        
        It's like me posting a Price is Right like game and having players guessing 4 grocery food item prices, if they guess the price is within 50c, they get it right. If all 4 prices are right, they win a car. Now if I added host actions, "the host will ALWAYS add an item that is slightly more expensive than it looks", while that information is nice to know, it's not needed, and you can bet the host will never tell that to the player.
        
        aragornelessar on 12/02/2022 - 00:16
        
        @hasher22:
        
        Let's say you're playing the game right now in the studio, you're not going to be told about every detail in the game such as "The host always reveals a goat and always offers a switch". You're going to step up to the host and play through my steps 1 through 3. The only information before step 1 is
        that there is a car behind 1 door and goats behind 2 doors, and for you to select a door with the car to win, then step 1 will begin.
        
        Yes, they won't give you the rules, but without them you can't come an definitive solution that there is one correct answer.
        
        my post will still state the same question at the end (to switch?) and the same probability will still occur whether or not its the Monty Hall problem or not.
        
        The wiki link I posted earlier (assuming it is right because I'm not a mathematician) says there are different results depending on different situations. Reading your step 1 to 3 (Monty Hall title or not), you could get the following situations - the first is your version because that's what you believed without stating it in the original post.
        
        The host acts as noted in the specific (vos Savant) version of the problem -> Switching wins the car two-thirds of the time
        
        The host always reveals a goat and always offers a switch. If he has a choice, he chooses the leftmost goat with probability p (which may depend on the player's initial choice) and the rightmost door with probability q = 1 − p -> If the host opens the rightmost door, switching wins with probability 1/(1+q)
        
        "Monty Fall" or "Ignorant Monty": The host does not know what lies behind the doors, and opens one at random that happens not to reveal the car
        -> Switching wins the car half of the time
        
        The host knows what lies behind the doors, and (before the player's choice) chooses at random which goat to reveal. He offers the option to switch only when the player's choice happens to differ from his -> Switching wins the car half of the time
        
        The host opens a door and makes the offer to switch 100% of the time if the contestant initially picked the car, and 50% the time otherwise -> Switching wins 1/2 the time at the Nash equilibrium
        
        There are other situations but I didn't repost the ones that did not involve Monty opening a door
        
        hasher22 on 12/02/2022 - 02:01
        
        -2
        
        @aragornelessar: Your examples are dissecting if scenarios. This is not related to the original complaint of me not mentioning Monty will always open the non-players goat door and always ask the player to switch.
        
        Point 1: Well that's the game
        Point 2: Well that's the game in the first sentence, don't even want to try to dissect whatever the rest is.
        Point 3 and 4: Pointless and irrelevant, why would Monty reveal a door before the player chooses? That's not how the game works nor what my question is asking.
        Point 5: Irrelevant as the final question is purely if the player wants to switch and people will based their decision of that. Whether or not the contestant chooses the car door and the host knows it, this is not relevant information to anyone.
        
        I will put it to you this way, this post is making the reader be the player and play the game. This is NOT a post where I am asking people to dissect the problem as your wiki does
        
        I will copy and paste my answer to you as well from another comment:
        
        Interesting, cause multiple different forum users asked the same game scenario without mentioning Monty will always reveal a goat door and yet the replies and answers are very similar to this thread.
        
        This is just on page 1 and 2 of my google search "Monty Hall Problem forums". This does not include forum links that ask for clarity of the problem or help with the problem. These are posts similar to mine asking the reader to play.
        
        https://www.bluegartr.com/threads/44052-The-Monty-Hall-Probl…
        https://forums.online-go.com/t/the-monty-hall-problem/35638?…
        https://forum.cyclinguk.org/viewtopic.php?t=148740
        https://gmatclub.com/forum/switch-or-stay-monty-hall-problem…
        https://www.mtgsalvation.com/forums/community-forums/talk-an…
        https://forum.philosophynow.org/viewtopic.php?t=8463
        https://aha.betterexplained.com/t/understanding-the-monty-ha… (asks the question to play and then explains the answer without stating Monty will always reveal a goat door)
        
        goedkoop on 12/02/2022 - 04:45
        
        @hasher22:
        
        I will put it to you this way, this post is making the reader be the player and play the game. This is NOT a post where I am asking people to dissect the problem as your wiki does
        
        Fair enough, as long as you agree there's no solution then. Depending on what assumption we make for how Monty is behaving, what we should do changes. It isn't unambiguously better to switch since we don't know what rules Monty is following. Here's a few different assumptions one might make about how Monty is behaving which are all consistent what what you've described, and how they would each induce a different response:
        
        Assumption 1: Whenever this game is played, Monty always reveals a goat and always offers a switch.
        
        "Switching gives me 2/3 probability of winning the car, while staying gives me 1/3. I should switch."
        
        Assumption 2: Whenever this game is played, Monty randomly chooses one of the two unopened doors to open (meaning sometimes he can reveal a car).
        
        "He has opened a goat in this instance only by chance - both switching and staying give me a 50% chance of winning the car."
        
        Assumption 3: Monty only offers switches when you have correctly picked the car - he is trying to save money for the producers of the show!
        
        "Since he is offering me a switch, I must have correctly picked the car initially. Switching gives me a 0% chance of winning the car."
        
        So on and so forth. In my opinion, one of the reasons this problem was so controversial when it was initially published was specifically because it was somewhat ambiguous how Monty operated. Many readers had different interpretations, and argued accordingly. I think there would have been a lot less controversy if the initial problem explicitly spelled out that he is pre-committed into revealing a goat and offering a switch. Anyway I'll stop arguing now, I think we're on the same page mate.
        
        Qazxswec on 12/02/2022 - 13:48
        
        @goedkoop: He is not asking you to make assumptions, he is telling you the facts, fact 1) you picked door 3, fact 2) he showed you a goat behind 2, those are the facts of the scenario he is asking, you can't change those facts because if you do, you are no longer answering the question he asked you, now that he has established fact 1 and 2, he is asking what you will do, stay or switch
        
        aragornelessar on 12/02/2022 - 13:31
        
        @hasher22: I clicked through the links that worked above (probably broken when you copied and pasted it from another post) and the fact that the titles are all named "The Monty Hall Problem" in the posts already biases the people who have heard of the problem and know the solution where switching is the better option (according to the standard rules). Just like other people have pointed out, your post doesn't specify these rules. Even the original author of the question in Parade magazine states it -
        
        "So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question."
        
        People are answering different questions to your intention because you didn't define those conditions
        
        I will put it to you this way, this post is making the reader be the player and play the game. This is NOT a post where I am asking people to dissect the problem as your wiki does
        
        Then you need to accept that there are different solutions that people are going to arrive at and the wiki also shows you these different solutions. You can't go around telling people they are wrong simply because they used different assumptions to you when you didn't explicitly states those conditions. If you had stated the standard conditions then that's a different story.
        
        Qazxswec on 12/02/2022 - 13:44
        
        -2
        
        @aragornelessar: but he does define the rules, he explicitly says, you pick door 3, the host shows you a goat behind door 2, those are the facts behind the scenario he is asking you, he is not asking you people to make assumptions about whether Monty always picks the goat, he is telling you straight up, Monty has picked the goat, then asking you if you will switch, not to mention the fact that anyone who knows the problem, knows Monty always picks the goat, but by his scenario alone with no other source of information, he picks the goat everytime, you cant change the OP's scenario and say what if he didnt pick the goat, that is not the question he is asking, i can understand why people are trying to dissect it and yes maybe it would of been a little clearer if he specifically said that Monty always picks the goat, but he tells you Monty picks the goat, you cant change that, if you do, you are changing the whole scenario and going off topic, no longer asking the same question as what the OP has asked
        
        goedkoop on 12/02/2022 - 14:28
        
        +1
        
        @Qazxswec:
        
        He is not asking you to make assumptions, he is telling you the facts
        
        We have explained it so many times above and you still aren't getting this. Knowing what Monty has done in this one instance is consistent with many possible behaviors. Different behaviors = different answers. Go read the Wikipedia page to see all the different answers there are when Monty's rules are left ambiguous.
        
        If OP just said "the rules of the gameshow are that Monty will always reveal a goat behind one of the other two doors and then give the player a chance to switch", then Monty's behavior is clear, there is no ambiguity and only one solution. Instead if you just say "today he revealed a goat and is offering you a switch", then I NEED to make an assumption about how Monty behaves in order to calculate any probabilities. So there's no one solution.
        
        Qazxswec on 12/02/2022 - 14:40
        
        -1
        
        @goedkoop: NO, you dont need to assume how Monty behaves on other days of the week, you have been told how he behaved in this specific example, you are the one not getting it, he is telling you how Monty has BEHAVED, keyword, not behaves, but behaved, in this specific situation, he picked the goat, the Op is asking you in the scenario where Monty picked the goat, what will you do next, stop trying to over analyse it and think to yourself "well maybe Monty doesnt always pick the goat", that is a different question, not the one OP is asking, by trying to assume whether he picks the goat or not every time and whether or not he behaves the same every time is irrelevant, you are changing the 2nd FACT, and that fact is that he picked the goat.
        
        goedkoop on 12/02/2022 - 15:14
        
        @Qazxswec: I know Monty picked the goat. This by itself isn't enough information to calculate the probability of winning from switching.
        
        Qazxswec on 12/02/2022 - 15:23
        
        -1
        
        @goedkoop: yes it is, Monty picked the goat, that leaves 1 car and 1 goat, there is a 2 in 3 chance you have a goat, switch, sorry, i know what you are trying to say, and yes it would of left speculation out for everyone if OP specifically said he always picks the goat, but the truth is, everyone these days try to redefine the parameters of specific questions people are asking, instead of following specific instructions, i am guilty of it too, i told someone in another post who was asking if blue filter glasses make you sleep better that they were not achieving enough during their day so they sub consciously didn't want to let the day go say they stayed up scrolling till late at night, that is some major douchebag shit right there, maybe its true, maybe its not, but it was definitely not the very specific question they were asking
OzzyOzbourne on 09/02/2022 - 10:36

I don't understand why you would switch. At that point in time, you have a choice of your door or the other door, 1 has a goat, 1 has a car. 50:50 chance. Doesn't matter what the odds at the start are or the door with a goat that's been revealed, those are the odds and choice now. So I agree with the phd holders.
- DiscountForThee on 09/02/2022 - 12:35
  
  Did you read any of the dozens of comments which explain why it isn't 50/50?
- goedkoop on 10/02/2022 - 04:34
  
  +1
  
  OP has told it poorly, so I don't blame you. The critical point is that Monty always reveals a goat door no matter what you've chosen. Since he is committed to always revealing a goat, switchers win the car 2/3 of the time (they win if they initially pick a goat), while non-switches win the car 1/3 the time (they win if they initially pick the car).
  - hasher22 on 11/02/2022 - 22:57
    
    Interesting, cause multiple different forum users asked the same game scenario without mentioning Monty will always reveal a goat door and yet the replies and answers are very similar to this thread.
    
    This is just on page 1 and 2 of my google search "Monty Hall Problem forums". This does not include forum links that ask for clarity of the problem or help with the problem. These are posts similar to mine asking the reader to play.
    
    https://www.bluegartr.com/threads/44052-The-Monty-Hall-Probl…
    https://forums.online-go.com/t/the-monty-hall-problem/35638?…
    https://forum.cyclinguk.org/viewtopic.php?t=148740
    https://gmatclub.com/forum/switch-or-stay-monty-hall-problem…
    https://www.mtgsalvation.com/forums/community-forums/talk-an…
    https://forum.philosophynow.org/viewtopic.php?t=8463
    https://aha.betterexplained.com/t/understanding-the-monty-ha… (asks the question to play and then explains the answer without stating Monty will always reveal a goat door)
    
    p.s i didn't neg you.
    - AnophthalmiaCervidae on 12/02/2022 - 11:33
      
      +1
      
      You post multiple links that are actually providing this additional information that was not presented in the OP question, the fact that Monty knows where the goat is before revealing it. Monty doesn’t just open a random door that in your case might just happen to reveal a goat.
      
      Premise: Gameshow host offers you the choice of three doors (A, B, and C)
      Behind two doors there is a goat (or undesirable prize)
      Behind the other door is a sports car (or desirable prize)
      You can pick whichever door you want.
      
      The gameshow host (who knows what is where) will then open exactly one of the other two doors, and show you a goat.
      
      **Goat reveal is not random
      
      The contest host Monty Hall shows the contestant Joe three closed doors. He tells Joe that behind one of the doors is a valuable prize. The prize goes to Joe if Joe chooses the door with the prize. Joe doesn't know which door holds the prize. He chooses one of the doors, maybe by some random process or perhaps by whim. Call his choice door A. Monty then opens up one of the remaining two doors — the one not holding the prize (Monty knows which one — call this door B). Replace the previous sentence by: "Monty, who knows where the prize is, then opens up one of the remaining two doors — one not holding the prize — call this door B". Then Monty asks Joe to stick with the original choice (A) or make a switch to the third door (C).
      
      Again
      
      There are 3 doors, behind which are two goats and a car.
      You pick a door (call it door A). You’re hoping for the car of course.
      Monty Hall, the game show host, examines the other doors (B & C) and opens one with a goat. (If both doors have goats, he picks randomly.)
      - hasher22 on 12/02/2022 - 16:09
        
        -2
        
        @AnophthalmiaCervidae: You don't understand the initial complaint. The initial complaint was from geodkoop: "Monty is committed to always revealing a goat behind one of the other two doors no matter which door you select as the contestant."
        
        The links I have provided DOES NOT mention Monty always opening a goat door.
        
        You're adding extra information to the argument which is irrelevant.
        
        My point is that those links does NOT mention Monty ALWAYS opening a goat door.
        
        Geodkoop made it clear that his argument was to mention the word "always".
        
        AnophthalmiaCervidae on 12/02/2022 - 16:12
        
        @hasher22: Did you read them?
        
        hasher22 on 12/02/2022 - 16:22
        
        -1
        
        @AnophthalmiaCervidae: Yup you quote one of the links where it state Monty will ALWAYS reveal a goat door is mentioned?
        
        Also, stating that Monty will always open a goat door is a given action in each game. There is no point mentioning it cause its part of the game. Knowing this information pre-game, will not benefit the player as it's part of the process which leads to the question to switch. Without opening a goat door, this problem would not exist. A player picks a door, the non-selected goat door opens either way, the game does not have a version to not open a goat door from Monty.
        
        Let me put it to you this way, the post is to make the reader play the game NOT to dissect and analyse the problem on a deeper level.
        
        AnophthalmiaCervidae on 12/02/2022 - 16:36
        
        @hasher22: Does not necessarily use the word “always” but they all say Monty knows where the goat is and opens a door with a goat.
        It is not random.
        In your description he just opens a door and it contains a goat.
        In fact your description says “You have a 33% chance to win the car”.
        You have a 33% chance of winning the car (whether you switch or not) if Monty’s choice is totally random.
        
        If it is random 1/3 of time Monty will reveal the car and you lose.
        If Monty reveals a goat (which he will 2/3 of the time) whether you switch or not it is 50/50 chance of winning.
        2/3 x 1/2 = 1/3 chance of winning.
        
        hasher22 on 12/02/2022 - 16:45
        
        -2
        
        @AnophthalmiaCervidae: My post in step 2, Monty opens the goat door, 2, as the player selected door 3.
        
        This already gave the player that Monty will not reveal the car door or the players selected door.
        
        My post is just for this game. When comments are suggesting stuff like, the host knows which door has what or the host will always open a goat door is irrelevant when playing the game.
        
        So what if the host knows what is behind each door or always reveal the not selected player's goat door. You as the player don't need that information to play the game. That's my point.
        
        AnophthalmiaCervidae on 12/02/2022 - 17:03
        
        @hasher22:
        
        This already gave the player that Monty will not reveal the car door or the players selected door.
        
        No it only gives that for this one instance of the game that Monty has opened a door with a goat, not necessarily that this will always occur every single time.
        
        That is what determines whether you are better off switching or not.
        
        hasher22 on 12/02/2022 - 17:09
        
        -2
        
        @AnophthalmiaCervidae: But that's legitimately the game, he will 'always' open a goat door (which is not the players door).
        
        There will never be a time where Monty will not open a door. Monty will never open the car door (if not selected) or the players door. It will always be the non-selected goat door, each and every game.
        
        That's why I'm saying that making that statement of Monty will always open a goat door is not crucial information to play the game.
        
        aragornelessar on 12/02/2022 - 16:37
        
        @hasher22:
        
        Let me put it to you this way, the post is to make the reader play the game NOT to dissect and analyse the problem on a deeper level.
        
        People have played the game as they've read it, gave you a 50-50 answer but then you're telling them they're wrong. How is that not analysing the problem?
- AnophthalmiaCervidae on 10/02/2022 - 09:45
  
  +2
  
  You buy a lotto ticket.
  I buy millions of lotto tickets covering every other possible combination but yours.
  One of us has the winning ticket (and only one winning ticket).
  After the lotto draw I check my tickets, setting aside the winning ticket if I have it, and show you losing tickets one at time until I only have one ticket left.
  2 tickets are left, one of which is the winner.
  Do you think that you then have a 50/50 chance of winning?
  Without having checked your lotto ticket would you keep your original ticket or swap it for mine?
clippers on 09/02/2022 - 11:30

+2

If you know the host always opens a goat door no matter what you chose, you improve your chances of winning car by switching.

If you know the host always opens at random no matter what you chose, you do not worsen your chances of winning car by switching.

But if the host only opens a goat door when he knows you have chosen the car, the game is rigged - shove the goat up the host!
truebunny on 09/02/2022 - 11:42

Switch

Easy explanation - the host is forced to open a non car door out of two options he has. Sure the other door he didn't open could have a goat too, but it is about winning in the long run. Meaning if this exercise is done thousands of times, it increases your chance of winning compared to if you didn't switch.
AnophthalmiaCervidae on 12/02/2022 - 18:28

Ok I was wrong.

Based purely on the information as given in the flawed original post, without assuming any pre knowledge of how “Lets Make a Deal” or the traditional “Monty Hall Problem” is supposed to work, and just “playing the game” as it was posed by the OP I would not switch.

The Original Post tells us

You have a 33% chance to win the car and a 66% chance to select a door with a goat.

33% chance to WIN the car (rounding forgiven), not a 33% chance it is behind your initial door 3 selection, or that your chance of winning may increase with the option to switch. You only have a 33% chance to actually WIN the car.

This then tells us that Monty’s choice of door was in fact actually random and that only in this particular single instance of the OP’s game show has he just happened to reveal a goat.

In a random choice 1/3 of time Monty will have revealed the car, you will lose and only have the option of switching one goat for the other.
If Monty has randomly revealed a goat (which he will 2/3 of the time) whether you switch or not it is then a 50/50 chance of winning.
2/3 x 1/2 = 1/3 chance of winning (approximately 33%).

This is the only behaviour from Monty that only gives us a 33% chance of winning. If Monty knows where the goat is and always reveals a goat your chance of winning would increase if you switch, but with the question as posted we know there was only ever a “33% chance to win the car”.

So down to a 50/50 choice, with the same odds whether I switch or not I would not switch. I would FEEL worse if I gave up a car behind the door that I had won than if I did not win a car behind the door that I never had.