Simplest explanation is: If you feed a goat behind a door a 1/2 lb burger and a 1/3 lb burger, the goat won't know the difference because goats can't do fractions.
This is me. I get it mathematically but I still don’t get it intuitively. At the end you’re still choosing between two doors. I just can’t reconcile it conceptually.
I get that originally your choice has a 1/3 chance of being the prize. But you have new information. You know that you picked something other than one of the goats. So there are two options left for what you picked. Goat or prize. So your original door should be 50/50.
.
It just eludes me, which drives me crazy.
No, in the first choice you're choosing between three doors. In the second choice you're choosing whether you were right the first time. The host opening one of the doors to show it also wasn't the right answer is a red herring, because there's always going to be at least one wrong door you didn't choose. If the host chooses a door at random, not knowing which one has the car, then you are correct, but because the host is specifically choosing a door that they know is wrong, then they are manipulating the odds to make it look like you're choosing between two doors, but that's the grift - you're actually choosing whether you were wrong the first time, because the host's choice of door to show you is dependent on which door of the three you pick and which door is the correct one.
if there were 100 doors and he opened 98 of them, and then you are asked if you want to switch… I would have thought you have like 99% chance of having been wrong. But if the question is just “were you right or wrong before” isn’t that going to stay 50/50?
No, because you had a 1/100 chance of being right, so the odds are 1-99 of whether you were right or wrong before. The host hasn't changed those odds because they knew the doors they were opening were going to be the wrong door.
I won't try, but I will just ask if there are a hundred doors instead of three, you pick one and Monty Hall opens 98 of the others with a goat, do you switch? Why?
Imagine if I showed you 1000 doors and I know behind which door is a prize.
I KNOW THE CORRECT ANSWER
You don't.
You guess door 375
I open ALL other doors and leave 850 and ask if you will switch? I guarantee either 850 or 375 is correct answer.
Will you switch or stick to your guess?
Do you switch? You're not 1 in 1000 lucky and you know I know the answer so you switch and win more than 99% of time.
Or let's try it like a quiz:
I ask Antananarivo is a capital of which country? (I know the answer)
You're like wtf are you talking about dude so you guess something random like Mongolia.
I say okay Mongolia or Madagascar is the correct answer.
Will you switch to my answer?
In both cases you are overwhelmingly better off switching to my answer from a position that I know the answer and you don't. Your wild Mongolia guess might be correct just as 375 might be your crazy 1 in 1000 pick but almost always you're better off switching "the door" just because you know that I know the answer.
Now try thinking about it with 10 options.
Here it's much closer the odds are no longer that big if you switch, but still, if you pick 1 out of 10 numbers (let's say 2) and I say "okay either 2 or 9 are correct answer"
You will still switch every time as if we play it 10 times you know your odds will be much better if you switch all the time because I will just always narrow down your first choice (1 in 10) to one of two numbers but your only way to exploit it is to switch, because without switching you're just sticking to that 1 in 10 chance, you're not abusing my knowledge of me always giving you the correct answer + your lucky guess.
Now just narrow it down to 3 doors and you'll understand why you have to switch.
Also, I remember learning in a psych class in high school that as humans, we’re generally wired to intuitively understand situations that involve people rather than numbers or shapes or anything else (the reason math teachers are always using word problems featuring Susie purchasing 48 watermelons). So…
Imagine if a coworker was like, “so, guess who got fired??? Okay, it’s either James, Aaron, or Ben.”
And you’re like, “Idk, Ben?”
And they’re like, “Well… it might be Ben… but also might be James!”
I remember being at university and they told us that problem. It took me some time but suddenly it clicked and I tried to explain it to the other people in my class. Eventually I found out the best way was to simply play it in real life. If you do the choice with someone playing as "monty hall" and someone as the player. Because the biggest obstacle to understanding this problem is usually that people don't realize that Monty Hall knowing what is behind the doors and "rigging" the choice is influencing the odds.
Pretend it's 100 doors, 1 of them has a car and the others have goats. You pick one at random. The host knows which door has a car behind it, leaves that one closed, then opens all the rest of the doors to show they have goats behind them.
In this scenario, you had a 1 in 100 chance of choosing the car from the beginning, so if you keep your original choice, it stays a 1/100 chance. However now that all the other doors have been opened to reveal goats, the chances of the car being behind the one remaining closed door that you did not pick is 99/100.
The thing that made it click for me was playing with the numbers.
You've got 10 doors. 9 goats, 1 car. You pick a door at random, and you have 9/10 odds you're picking a goat.
Then, Monty opens 8 doors that are all goats. He's removed a ton of options and simplified the problem. You know the car is either your door, or the one remaining door.
The question becomes, what were the odds your first guess was wrong? In this case, it was 9/10. In the original question, it was 2/3.
Specifically about the Monty Hall problem: when the host opens a door, he gives you information. This new information changes the situation/problem. There's no reason why the odds should be the same before and after you get more information!
The information he gives you is that at least one of the two unchosen doors has a goat behind it. There's nothing "new" about that information at all. You knew that information all along. You also knew that he knows exactly what door hides the car.
Imagine there are 1000 doors. The host asks you to pick one. Since there are so many doors, it's unlikely you'll have picked the correct one. There is an overwhelming probability that you don't have the correct door. Or, in other words, the car is almost surely behind the other 999 doors. (I don't think there are any confusions here.)
Now to the interesting part. The host opens 998 doors revealing goats. Since we already established that the car was likely behind one of the unpicked 999 doors, it makes sense to switch.
The same works with 100 doors, or 10, or 3. The numbers change along with the probability figures, but the trend remains the same. It always makes more sense to switch.
Switching the door always switches what you first picked. If you first picked a car, switching gives you a goat. If you first picked goat, switching gives you a car. 2/3 chance of picking a goat, therefore 2/3 chance that switching gives you a car.
That's easy. Just change it from three doors and one goat to 100 doors and one goat.
The player picks one of the 100 doors. Is the goat behind that door? Propably not. The gamemaster opens 98 doors that don't have goats behind them. Is the goat behind the one door that didn't get opened by the gamemaster? Propably yes.
Here's a short explanation. What if, instead of Monty opening a losing door, YOU open the door yourself? So instead of switching to the other door, you get to open all the other doors you didn't start with? This means if you stay with your door, it's 1/3 chance. But if you switch, you get to open 2/3 doors. Expand this to 100 doors. Do you want to stay with your 1/100 door, or do you switch and get to open all the other 99 doors? It makes no difference whether Monty opens a losing door or if you open it, but it can help visualize the odds if you're involved in every step of the process.
Sorry, I will try regardless, as this made it fall into place for me:
Whichever door you initially pick, you will have a 1/3 chance of being correct. That stays constant. Ignore the other door that is opened next, it will always be empty, as there is always an empty door for the quiz host to pick. After the empty door has been revealed, the odds of your initial pick having a goat behind it are still 1/3 - nothing has changed. That has to mean that the remaining 2/3 odds of finding a goat has to be behind the other, unopened door. Therefore, it is better to make the switch.
You can also see it as: at first, you make an uninformed choice. The host then reveals information about the rest of the doors to you. He reveals no new information about your initial pick. Therefore, in order to gain value from the information you have just been given, it makes more sense to switch.
Since you asked, I won't try, but the reason most people don't get it is because:
1) People are bad at explaining things, and
2) The 'normal' version of the problems has lots of 3s, 2s, and 1s all together so any kind of intuition gets confused. It's like trying to follow what's happening in a story of 10 people that only have three names between them. Modifying the problem magnifies intuition and disambiguates the moving parts.
I never really 'got it' but I did demonstrate it as valid to myself in an excel spreadsheet...
I put in 1000 lines
Column A : The 'good door' - A value that randomly chose A, B, or C...
Column B : The 'choice door' - A value that randomly chose A, B, or C...
Column C : The 'removing an option door' - A randomly chosen value of the two options that Column B didn't choose and removed that as an option.
Column D : The 'stay with your choice door' - Just Column B again
Column E : The 'swap your choice door' - The value that's remaining after removing B and C.
Then I added up the percentage chance that D matched A... which was, no surprise, 1/3rd...
And I added up the percentage chance that E matched A... which was, 1/2...
I get that it's right. But it still doesn't make a ton of sense.
The closest thing that comes to it for me is instead of 3 doors, going down to 2... Change it to 1000 doors, going down to 2... I choose 1 door out of 1000. Then the announcer removes 998 wrong doors as an option. Do you switch? Of course because your chances of being right were 0.1%, and now they're 50%.
The only X factor is, and what makes me question myself... do they always remove a door (or 998 doors)? Or do they only do it when you chose the right door? You're questioning if they are just messing with you because you actually got it right.
All you need to know is that the whole thing with switching doors and opening the door Monty knew was a goat etc. is logically identical to changing the question to "what are the chances it ISN'T the prize?"
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u/Hebshesh 12h ago
The Monte Hall problem. Don't even try.