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 âswapâ door because by the time you actually âswapâ, 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 âold knowledgeâ.
Thatâs the way it fits my mind from a quantum, informational perspective.
âSince your original decision was based on (2) goat variables, it now makes more sense to switch now that there is only (1) goat variable.â
Iâm 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 thingâŠWhich of the other two doors do you switch to? After all, whatâs 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 donât switch from a goat to the other goat. Now if you switch, youre choosing between two different decision treesâŠ
One when two goats existed as variables, and the more recent, updated timeline where only one exists. Youâre 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 instantlyâŠThe only thing Monty does is show you which door to switch toâŠkeeping 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 doesnât 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 canât do it until he keeps you from picking another goat.
May I remind you, double-0 Nobody, that this blog is entitled âBetter Explainedâ. In that regard, long-winded, confusing and, not to mention, incorrect explanations and statements such as:
âMonty fixes that scenario of switching from a goat to a goat by opening one door and this ENSURES you donât switch from a goat to the other goat.â
If you were more likely to be wrong the first time, then tell me why you wouldnât switch immediately?
The only reason could be because you donât want to switch to another goat, and you dont have information about which door it is behind. Thatâs the only thing that changes.
More importantlyâŠjust because you donât 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)
âHow does fewer goat variables point to one door over another?â
The answer I was looking for was: âYour 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 variablesâ. Yes, that short.
And Montyâs 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.
Please can you comment on our reasoning? I know youâre tired of explaining the same thing repeatedly; Does the probability of the ânot-my-choiceâ door change to 2/3 when monty reveals a goat, or is that a simplified way of explaining probability to unlearned people like me?
âDoes the probability of the ânot-my-choiceâ door change to 2/3 when monty reveals a goat?â 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 itâs equally likely heâll open Door3 instead)
So, when you see Monty open Door2 you can deduce heâs twice as likely to do that when the car is behind Door3 than when itâs 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 Itâs 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 youâve seen that one of the other doors doesnât have the car behind it or not doesnât 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 didnât pick at first represent a 2/3 chance of hiding the car. Finding out that one of those doesnât have a car behind it â which is a given whether Monty Hall opens the door to show you or not â doesnât change the probabilities.
Explaining it a different way: You know from the outset that of the two doors you donât pick initially one of them doesnât have the car behind it (thereâs just one car), so Monty Hall opening that door reveals nothing you didnât already know. He is never going to reveal whatâs 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 didnât open increases to 2/3.
You donât actually get to trade your 1 door for the 2 other doors, if that were the case there wouldnât be any point in Monty opening a door at all = and youâd end up with either 2 goats or 1 goat and the car
@PalmerEldritch â There is no point in Monty Hall revealing the goat. Thatâs just window dressing. You know going into the game that one of the two doors you donât pick hides a goat, and you know that Monty Hall will ALWAYS show you the goat. You learn nothing from the reveal. The host isnât choosing a door at random to reveal â he always show whatâs behind a losing door. Revealing the goat is a misdirection and suspense-builder, not new information.
If the host didnât open a door to show you the goat but you still got the chance to trade your one door for the other two itâs obvious that trading is the best strategy. Since you donât 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 donât have time to read all of them.
Iâd 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 Iâd 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.
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 instantlyâŠthis is a given. The new information is âdont switch to this other goat door!â He gives you (1) âswitchâ possibility instead of (2).
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 choiceâŠi.e. which door do i switch to? Heâs basically saying "Youâre positive you want to switch, but you donât 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 âswitchâ door.
@Nemo â Youâre 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 thereâs no advantage to keeping your first choice or switching.
Seeing the goat is not relevant â you already know that one of the remaining doors hides a goat, and you know that the host will show it to you. He isnât eliminating a choice because you are not offered the chance to choose either one of the other doors you didnât pick â that isnât how the game is played. You are given the chance to switch to the door the host didnât open.
Maybe weâre saying the same thing. My point is that seeing the goat doesnât 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 didnât pick the car on your first try, not because you saw the goat that you already knew was behind one of the doors you didnât 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) - âOh wait a second, at a 2/3 chance that means I probably picked a goat so I should switch!â
âOk, so should I switch to B, or C⊠Iâd hate to get a goat âŠhmmm I cant really switch right nowâ
Monty: Hey contestant, Iâm going to make the game interesting and reveal one of the doors. - OPENS DOOR
âOh, 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!â
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 youâd still have that 2/3 chance of getting a goat on the first choice.
He doesnât 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 wouldnât 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.