User talk:Coffee2theorems/MHP

Material related to the Monty Hall Problem (MHP) controversy.

MHP interpretations

Problems like this use a sort of code, a puzzle-lover / probabilist jargon, that signals to you how to interpret it. People have lots of different ways of doing that in general. In one way or another, you form in your mind a Platonic ideal ("essence") of the problem, and declare that anything not matching it is a different problem. People may have different Platonic ideals given the same problem statement.

If you blindly pull a marble out of a bag with two blue marbles and a red marble wrapped in opaque candy wrappers in it, you do not condition on it being the left/middle/right marble from your standpoint, even though you know the difference by feeling it. Why? Jargon. Convention. Other reasons. If the marbles are arranged in a row on a table, many do condition on location. Why? Jargon. Convention. Other reasons. Putting a bag around the marbles does not magically make them indistinguishable, yet they are generally assumed to be so. Or one may argue that the marbles in a bag are assumed to be "perfectly jumbled" in some sense, and marbles on a table are not; either way, the bag signals to you that you should assume lack of knowledge or even "true" randomness. That real marbles may not be sufficiently jumbled is considered irrelevant; similarly, when a deck of cards is shuffled, it's assumed to be in a "perfectly random order", despite real shuffles being very far from perfectly random. What happens with marbles on a table is more controversial.

Some interpretational issues of "the MHP" (many from actual discussions, some taken by analogy from actual discussions of other probability problems, some obtained by taking an approach to its logical conclusion) listed below:

The/any MHP statement is to be read as a puzzle / word problem using puzzle-lover / probabilist jargon, as follows:

The use of "a door" (unidentified door) is jargon signaling that door identity is not to be conditioned on.
The use of "door 3" (identified door) is jargon signaling that door identity is to be conditioned on.
The phrase "chooses a door, say, door 3" is jargon that means the same as saying...

..."chooses a door" (unidentified door, don't condition on it!).
..."chooses door 3" (identified door, do condition on it!), and implies that number 3 is special.
..."chooses door 3" (identified door, do condition on it!), but implies that the problem remains the same if it is retold using a different door labeling scheme; number 3 isn't special.

The phrase "you're given the choice of three doors" is like saying...

...you blindly pull a marble out of a bag with three marbles in it.
...you pick a marble off a table with leftmost, middle and rightmost marble on it, each covered by an opaque shell.
...hey, these options aren't any different! The marbles in a bag are also spatially distinct!
...no, anyone making that objection misunderstands the jargon!

The phrase "opens another door, which has a goat" is like saying the host reveals...

...a sock.
...a shoe.
...the left shoe of a pair, with the right shoe still hidden.
...hey, some of the previous options aren't any different from each other!
...no, they are all different, and there are even more subtle distinctions not listed!

An unvarnished "probability" is asked for, not a probability given something (the keyword given is missing!). This implies...

...nothing. There are other ways of signaling intent. In this case, depending on your interpretation of "..., say, ...", you should either condition on "a door" or "door 3" being open.
...that unconditional probability is meant. This is always so when the keyword given is missing. Any trappings suggesting conditional probability are red herrings; it's a common type of trick question testing whether you've got your jargon down pat.
...wait a minute! The question is whether you should switch; a decision is asked for, not a probability!
...but the probability is an essential part of the problem! If it's 50:50, you can argue you should still switch, as it isn't any worse! The point is that there is a big difference in the probability!
...and you still aren't asked whether you should switch given that you are in the situation with door 3 open!

The/any MHP statement is to be read in the context of a Bayesian world-view and the Bayesian way of talking about probability, as follows:

You are in a decision-making situation with one door open. The way to make a decision is to compute probabilities conditional on all you know (or an expectation conditional on all you know, if you have a decision theoretic bent).
"All you know" is nice rhetoric, but in practice, you do not condition on e.g. Monty's demeanor, even though it can be hard to miss and is probably informative. You can't take "all you know" at face value, and you need to define what it means. "All you know" includes...

...whatever you think the MHP statement says you know when read using your favorite brand of probabilist jargon.
...whatever is known in a simplified model you feel the problem is equivalent to, or retains what is (to you) the essence of the MHP; no violence is done to your Platonic ideal of the MHP by the simplification. This could have...

...Monty using a big screen instead of real doors, with pixel-by-pixel indistinguishable closed-door images and likewise indistinguishable opened-goat-door images.
...further, instead of actually opening a door, Monty could write something identifying the remaining door on paper and seal it in an envelope handed to you, and ask you whether you want to switch to the door named in the envelope.
...or you could have two blue marbles (goats) and one red marble (car) in a bag; the essential paradox remains.

...a door and a goat. You feel it reasonable to assume the player doesn't know which door was opened, just like you assume you don't know which marble you pulled out of a bag, despite them almost surely being spatially distinct (e.g. leftmost, middle, rightmost marble in the bag from your POV) and distinguishable by touch.
...door x (some specific door) and a goat. You feel it reasonable only to assume the player doesn't know which goat was shown; the door is visible and hence clearly known (e.g. left/middle/right door), but a goat is just a goat to you and the player.
...door x and goat y. You feel it reasonable to assume the player knows which door was opened and which goat was shown.
...door location x and e.g. goat size y (say, y times the average goat). You feel you should condition only on something that is definitely observable, and that the player ought to know both opened door location and revealed goat size/gender/etc.
...door location x. You feel you can ignore goat size/gender but not door location due to large observational uncertainty in goat size/gender but complete certainty in door location. Any model is an approximation, there is no such thing as "the correct model" or "the correct answer" to a vague problem, and this level of approximation is good enough for government work.

When people follow a particular way of interpreting problems and assume others do so also, it results in confusion and conflict. When they further insist that others must follow their way, for their way is the right way, it results in even more conflict. What is natural to one person may be completely unnatural to another. This is observed in other probability problems as well, such as the Three Prisoners problem, the Boy or Girl paradox and the Sleeping Beauty problem. -- Coffee2theorems (talk) 15:56, 7 October 2012 (UTC)[reply]