Understanding the Monty Hall Problem

The way I was able to understand it from a programming standpoint is that you start with:

3 variables = all doors are unknown contents

Then:

1 variable = becomes a constant (goat, and known location)

Originally that goat was a variable which affected your original choice. Now it no longer can affect your second choice.

Since your original decision was based on (2) goat variables, it now makes more sense to switch now that there is only (1) goat variable.

In essence, your choice is less affected by goat variables than it was previously.

I see it in terms of quantum, where originally as an observer all doors contain both goats and the car. Then, the state of one door/goat becomes known. This goat only affected your original choice, but never affected the “swap” door because by the time you actually “swap”, that goat will have a fixed location, which in this case is simply the knowledge of it not being in the door you swap to. Thus you are now competing against only (1) goat, not (2) as you originally were.

Put simply: The goat variable which affected your choice earlier in time is no longer able to affect your choice in the future.

It shows that we should always re-evaluate our actions and decisions after gaining new information or learning, and not to follow ideas based on “old knowledge”.

That’s the way it fits my mind from a quantum, informational perspective.

  1. ‘Since your original decision was based on (2) goat variables, it now makes more sense to switch now that there is only (1) goat variable.’

I’m not sure how the quantum explanation informs the decision to switch. It does not seem to offer any preference for one door over the other. Why switch? Why not toss a coin between the two remaining doors? How does fewer goat variables point to one door over another?

At the time you made the decision, you had more cons than pros. It actually would make more sense to switch instantly, except for one thing
Which of the other two doors do you switch to? After all, what’s the point of switching from a goat to a goat.

Monty fixes that scenario of switching from a goat to a goat by opening one door and this ensures you don’t switch from a goat to the other goat. Now if you switch, youre choosing between two different decision trees


One when two goats existed as variables, and the more recent, updated timeline where only one exists. You’re really only betting on the fact that your initial choice was more likely to be goat than car just based on quantities of each.

Thus you should switch instantly
The only thing Monty does is show you which door to switch to
keeping you from being in the exact same position as you were previously (with 2 goat variables)

So its as if you arrived on the scene knowing your first guess is 2/3 likely to be wrong and tell him immediately that you want to switch because of that fact. But switching without knowing where at least 1 goat lies doesn’t help your chances for fear of ending up with the second goat, so he reveals the door you should not switch to.

In other words, switching is already indicated from the outset of the problem, just due to the quantities of goats vs cars. But you can’t do it until he keeps you from picking another goat.

The two things are intertwined.

May I remind you, double-0 Nobody, that this blog is entitled “Better Explained”. In that regard, long-winded, confusing and, not to mention, incorrect explanations and statements such as:

“Monty fixes that scenario of switching from a goat to a goat by opening one door and this ENSURES you don’t switch from a goat to the other goat.”

may not assist.

Try not to muck it up next time.

M.

If you were more likely to be wrong the first time, then tell me why you wouldn’t switch immediately?

The only reason could be because you don’t want to switch to another goat, and you dont have information about which door it is behind. That’s the only thing that changes.

More importantly
just because you don’t understand something does not make it incorrect.

(Aiming for the shortest correct explanation)
Initially you pick #1 which is 1/3 correct. Nothing about what happens with revealing a goat provides any info. about door #1, so its still 1/3, and the remaining door must be 2/3 (i.e. 1 minus 1/3)

‘How does fewer goat variables point to one door over another?’

The answer I was looking for was: ‘Your first choice door had (and retains) two goat variables and one car variable, while your swap choice door has one goat variable and two car variables’. Yes, that short.

And Monty’s door has one goat constant. Altogether, this looks to me the same as saying the respective chances of car have changed from 1/3, 1/3 and 1/3 to 1/3, 2/3 and 0/3.

Those statements seem to make sense, especially the second one. That’s a good way of representing it mathematically.

Longwinded Brendan!

As first pick has 1/n chance, the only other door must have (n-1)/n chance, where n = number of doors in game initially.

PalmerEldritch

Please can you comment on our reasoning? I know you’re tired of explaining the same thing repeatedly; Does the probability of the ‘not-my-choice’ door change to 2/3 when monty reveals a goat, or is that a simplified way of explaining probability to unlearned people like me?

@Zenasdad

“Does the probability of the ‘not-my-choice’ door change to 2/3 when monty reveals a goat?” Yes it does. A non-mathematical explanation as to why is:

In the example where you pick Door1 (say) and Monty opens Door2 (say) then the question to ask is: why did Monty open Door2 (instead of Door3)?
If the car is behind Door3 then it is 100% guaranteed Monty will open Door2,
If the car is behind Door1 then it is only 50% guaranteed that Monty will open Door2 (as it’s equally likely he’ll open Door3 instead)
So, when you see Monty open Door2 you can deduce he’s twice as likely to do that when the car is behind Door3 than when it’s behind Door1, so switching to Door3 will give twice (or a 2/3) the chance, while staying with Door1 only gives a 1/3 chance.

PalmerEldritch

I totally grok that (780). If you had paypal I would send you $$ so you could go see John Wick or get a beer!!

Try to be more patient with uneducated cats such as myself.

@PalmerEldritch It’s logically equivalent. The goats and Monty Hall are canards. At first you get to pick one door. Then you get to trade for the other two if you want. Whether you’ve seen that one of the other doors doesn’t have the car behind it or not doesn’t change the fact that trading a 1/3 chance for a car for a 2/3 chance is a good trade.

It been explained in this thread numerous times that the two doors you didn’t pick at first represent a 2/3 chance of hiding the car. Finding out that one of those doesn’t have a car behind it – which is a given whether Monty Hall opens the door to show you or not – doesn’t change the probabilities.

Explaining it a different way: You know from the outset that of the two doors you don’t pick initially one of them doesn’t have the car behind it (there’s just one car), so Monty Hall opening that door reveals nothing you didn’t already know. He is never going to reveal what’s behind the door you chose at first, and he is never going to reveal the car. The important point is that you are then given the opportunity to trade your one door for the other two, trading your 1/3 chance for a 2/3 chance.

I agree the probability that the door you picked first remains at 1/3, the probability that the door Monty didn’t open increases to 2/3.

You don’t actually get to trade your 1 door for the 2 other doors, if that were the case there wouldn’t be any point in Monty opening a door at all = and you’d end up with either 2 goats or 1 goat and the car :slight_smile:

@PalmerEldritch – There is no point in Monty Hall revealing the goat. That’s just window dressing. You know going into the game that one of the two doors you don’t pick hides a goat, and you know that Monty Hall will ALWAYS show you the goat. You learn nothing from the reveal. The host isn’t choosing a door at random to reveal – he always show what’s behind a losing door. Revealing the goat is a misdirection and suspense-builder, not new information.

If the host didn’t open a door to show you the goat but you still got the chance to trade your one door for the other two it’s obvious that trading is the best strategy. Since you don’t learn any new information when the host shows the goat that should have nothing to do with your decision to switch.

I see my argument was made as far back as comment #80. The comments are entertaining, to say the least, though I don’t have time to read all of them.

I’d be curious to know how many actual contestants stuck with their original pick and how many switched. If half stayed and half switched the show would be giving away about 50 cars for every 100 contestants: 33 for every 100 who keep their first pick, and 67 for every 100 who switch, so 100 cars for every 200 contestants. If I were the producer I’d have to plan to give away cars to half of the contestants, on average, unless I knew that most of the contestants were either ignorant or fluent with probabilities.

That’s only partially true Greg


Think about it for a minute.

You are correct that you should switch immediately. But without eliminating one of your options, you would always be in the exact same position upon arrival.

You want to switch instantly
this is a given. The new information is “dont switch to this other goat door!” He gives you (1) “switch” possibility instead of (2).

If you arrive and were going to pick (A) and instantly decide to switch, you now have (B) and © to choose from switching to. But how do you know which is a better switch? When he opens © you know not to switch to ©, in fact you cant. When he opens (B), you know not to switch to (B).

Your switch option was a given at the beginning of the problem, just based on the odds of the game. The only thing you gained was (1) door to switch to instead of (2).

After deciding to switch (a given), it keeps you from having to make a second choice
i.e. which door do i switch to? He’s basically saying "You’re positive you want to switch, but you don’t want to switch to a goat so here let me show you where that goat is and you go ahead and take the other door as your “switch” door.

@Nemo – You’re describing a different game, where you pick door A, then are offered a choice to keep A or switch to either B or C. In that game there’s no advantage to keeping your first choice or switching.

Seeing the goat is not relevant – you already know that one of the remaining doors hides a goat, and you know that the host will show it to you. He isn’t eliminating a choice because you are not offered the chance to choose either one of the other doors you didn’t pick – that isn’t how the game is played. You are given the chance to switch to the door the host didn’t open.

Maybe we’re saying the same thing. My point is that seeing the goat doesn’t give any new information to you, so it should not influence your decision to switch. You switch because the odds are 2/3 that you didn’t pick the car on your first try, not because you saw the goat that you already knew was behind one of the doors you didn’t pick before you made your choice.

No, I only said that you already determined you had bad odds upon the start of the game. This means you should switch even when there are (3) doors. Imagine you have inner dialogue going on while everything is happening.

I pick (A) - “Oh wait a second, at a 2/3 chance that means I probably picked a goat so I should switch!”

“Ok, so should I switch to B, or C
 I’d hate to get a goat 
hmmm I cant really switch right now”

Monty: Hey contestant, I’m going to make the game interesting and reveal one of the doors. - OPENS DOOR

“Oh, wow the other goat is behind C! Well I originally wanted to switch seeing as how my first choice was more likely to be wrong. Now I have the chance to do it without repeating my first mistake!”

Moral of the story being that switching was always going to be smarter, but switching to another door before any variable changes would have you caught in an infinite loop because you would have to keep repeating that dialogue about being wrong on your first choice.

I think people are forgetting you never actually opened your first door. While staring at 3 doors you could switch all day in your mind and you’d still have that 2/3 chance of getting a goat on the first choice.

He doesn’t reveal any new information about what variables (object types) are in play, only rather he keeps you switching from a goat to a goat.

To illustrate this mentally, assume that Monty gives you the option to switch but without opening any door. Why wouldn’t you do this?

Only to prevent yourself from repeating your first mistake, which was choosing a door which was 2/3 likely to be a goat.