Monty Hall revisited, with a twist- survey

[DPCondit]

I fail to see the paradox.

I am not going to post formulas again, but just a simple explanation.

The difference between Monty Hall, and the Professor, is that Monty Hall can only choose from the two doors that you did not pick, this is a given. The Professor shows an answer that could have been any one of the three.

Therefore, and this should be clear enough by this time, if Monty Hall can only choose from answers that you did not pick (either by flipping a coin, or deliberately, this point is irrelevant, the only relevant point, is that he is not allowed to choose your pick), therefore, your answer still has exactly a one third chance of being correct. If he had chosen to expose this wrong choice without knowing which one that you had picked, then that changes everything, now he is exposing simply a wrong choice (either randomly or deliberately, once again, this point is irrelevant, the only relevance is that he has not excluded your answer from being exposed, and doesn't know what you picked), and therefore both remaining choices are affected equally, and are 50/50 shots.

So perhaps you could not understand the earlier explanations (hopefully you understand this one), but there simply is no paradox here. Just a problem that is somewhat counterintuitive upon first examination.

Don

[BruceZ]

In mathematics, the term "paradox" takes on a more specific meaning than it does in everyday language. A mathematical paradox is a fundamental contradiction which cannot be resolved by the theory in which it arises, and which thus shows that the theory is incomplete, and that the contradiction may only be resolved if a change were made to the theory, or if the theory were replaced by a new theory. A true paradox presents two plausible statements which cannot simultaneously be true, yet cannot be individually proven to be either true or false by the existing theory.

When a problem presents us with an intuitively obvious but actually wrong solution, it is NOT generally called a mathematical paradox. The wrong solution is simply termed a "fallacy". We also do not have a paradox simply because a problem offers no solution at all, on the basis of the stated conditions of the problem. If the problem cannot be resolved due to insufficient information, even though it is not immediately clear that extra information is required, the problem is termed "ill-posed". The term "conundrum" can also be used to describe this type of problem.

Sometimes the term "paradox" becomes attached to a problem which was once thought to be a paradox, but which was later found to be merely a fallacy, or perhaps a conundrum. For example, the so-called "twin paradox" offered to disprove Einstein's theory of relativity is actually just based on a fallacy, so it is not a true paradox. On the other hand, the "EPR paradox" offered by Einstein to disprove quantum theory could not be fully addressed without taking a particular interpretation of quantum theory which required additional experimentation to verify, hence it is a true paradox. Recent experiments offer strong validation for the proposed interpretation of quantum theory, but this specific interpretation was required to resolve the paradox nonetheless. This could be considered a resolved paradox; however, it is still not fully resolved to everyone's satisfaction. Schrödinger’s cat is also a true paradox which experimental evidence may soon resolve. Russell's paradox was a true paradox which imposed radical changes to logic and set theory. There are other paradoxes. True paradoxes are usually truly mind blowing, almost by definition.

The Monty Hall problem is far from a paradox. If we are not told how Monty chooses the door to show us, then the problem is ill-posed and cannot be answered without imposing our own assumptions. If we are told that a) Monty is constrained to show us a door that we did not choose, b) Monty must show us an incorrect door, and c) when we choose correctly, Monty chooses between the two incorrect doors to show us with equal probability, then the problem is not a paradox, and it is not ill-posed. It is completely and quite easily solvable. If someone were to argue after being given all this information that it shouldn't matter if we switch, then that person would be committing a fallacy, plain and simple. If we are told something different regarding conditions a, b, or c, then the problem can still be solved, but the answer may be different. Condition c is generally ignored by every discussion of this problem I have ever seen, yet it is absolutely essential information.

For examples of mathematical fallacies and paradoxes, and the definitions I have given here, see this entertaining and very readable text:

Mathematica Fallacies and Paradoxes

[BruceZ]

I should clarify something about the EPR paradox. I made it sound as though quantum theory needed to be modified in order to resolve this paradox. This was not the case. In fact, some clever math by a guy named Bell showed that the apparently impossible effect predicted by the paradox was real and could be derived directly from quantum theory. This paradox did not expose a contradiction within quantum theory, but an apparent contradiction between quantum theory and special relativity, both of which are intended to be parts of a larger consistent theory of the physical universe. Experiments have shown that quantum theory comes through with flying colors, and that the results predicted by quantum theory really do occur, even though these results are extremely bizarre, so much so that Einstein proposed the paradox because he felt that they cannot possibly occur. In fact, these results do not explicitly contradict the specific statements made by relativity. It is not understood, however, how such results are able to co-exist with relativity. No established theory exists which explains how such results can occur without violating relativity, and the ones that have been proposed are even more bizarre than the paradox itself, i.e. parallel universes, backwards time travel, reversal of cause and effect, etc.

I consider EPR to be the single most interesting topic in the universe, and I've given a lot of thought to bestowing that title upon it. It is highly relevant to probability, and I intend to post more on it when I get time to do it justice. The whole issue is not even known to most people because the true nature of quantum theory is usually obscured by sophisticated mathematics. Even those who study quantum theory as part of undergraduate physics or engineering courses never even come close to appreciating its bizarre implications, and this is partly because the implications are so bizarre as to strongly resist a correct interpretation. It is not widely known, for example, that this field has already produced what can accurately be described as teleportation on a small scale in a laboratory, and plans are underway to teleport atoms and even molecules the size of viruses. EPR also attracts a lunatic fringe interested in extrasensory perception and so forth, but what I am telling you is very real hard science.

In a nutshell, EPR stands for Einstein-Podolsky-Rosen, and the paradox predicts that certain types of particles must under certain conditions exhibit an instantaneous effect on each other, even if those particles are separated from each other by the vast expanses of the universe. Experimental evidence confirms that this happens. Relativity tells us that no energy or information can travel faster than the speed of light, therefore such instantaneous action at a distance would seem impossible. It turns out that no information has ever been observed traveling faster than the speed of light in these experiments because the result of these interactions can only be verified after the fact. Therefore, the particles seem to have found a loophole in the laws of relativity which allow them to perform this feat without being caught violating the theory. How this is possible is the big mystery, and is truly a paradox.

[Cyrus]

Elsewhere in this thread, a noted contributor to these pages makes the important distinction between a paradox in everyday, layman's terms and in mathematical terminology. Though not as qualified a mathematician as that noted contributor, I should probably clarify that the Monty Hall problem is indeed, as the contributor mentions, not a paradox in the mathematical sense : all the information necessary to solve the problem is clearly given to the solver and the problem has but one solution, i.e. always switching.

The Monty Hall problem, nonetheless, has entered everyday language also, if not mostly, as a paradox. A quick web search brings forth a plethora of sites that denote the problem as a paradox, and quite a lot of them are mathematical websites. Some of the texts point out that the problem is seemingly a paradox, others don't. (For instance, a paper out of MIT begins with the words : "The Monty Hall problem is based on an apparent paradox that is commonly misunderstood, even by mathematicians." Even professional mathematicians will often say that there is no difference between switching and non-switching, as the paper tellingly reports. In other words, mathematicians can be betrayed by their intuition --and not by their calculations-- which leads them to the wrong answer.)

The mathematical definition of a paradox is this : An argument is a set of statements (aka the premises) that leads to another statement (aka the conclusion) which follows from those statements. A paradox is an argument that involves a set of apparently true premises P1, ... , Pn and a further premise Q such as that we can derive a contradiction from both [P1 & ... & Pn & Q] and [P1 & ... & Pn & notQ]. The Monty Hall does not fit this definition.

Important clarification :

Mathematicians distinguish between logical paradoxes, such as the paradox by Russell, which was also mentioned by the distinguished contributor, and semantic paradoxes, such as the liar's, aka the Cretian's, well-known paradox. It is important to note that, although of course mathematics is a most robust science, if I may say so, semantic concepts creep into it unavoidably, despite the purists taking offense at such an infamy, as this is clearly demonstrated in those semantic paradoxes. (Other examples of semantic paradoxes are Berry's paradox, Grelling's paradox, Richard's paradox, etcetera). Russell's paradox which belongs in the former category and befuddled set theory, led to that theory coming forth with a better definition of sets, i.e. sets to be defined by their members rather than by general conditions. In other words, the premises were immproved in order to accommodate, and thus nullify, a mathematical paradox as such.

--Cyrus

PS : In my above post I also argued that the Monty Hall problem has nothing to do with philosophical conundrums. One could stretch our analysis of human perception to accomodate philosophical queries about "reality" and "denomination", the stuff paradoxes are made of, but that would be, as I said, stretching it : there is no philosophical aspect to the Monty Hall.

PPS : The poster who dissents to calling the Monty Hall a paradox (and, in mathematical terms, he is of course correct, if I may say so once more) also showed in his post examples of true mathematical paradoxes. In every single one of them, the scientists inform us that we will most probably, if not certainly, resolve those paradoxes when we are able to construct & carry out experimental verification and/or when our knowledge expands. (And this should include the EPR paradox, which in our contributor own's words "is the single most interesting topic in the universe".) In this sense, would we be correct to argue, only half in jest, that everything that, on the basis of the current totality of out knowledge, i.e. our current premises, we cannot understand or resolve, is a scientific paradox ? In other words, is it correct to say that everything we cannot yet understand is a temporary paradox? (But this would impertinently presume that eventually we will be able to understand everything! I am sorry, Father, I'll take 20 Hail Marys and fast. [img]/forums/images/icons/frown.gif[/img] )

PPPS : Although this may come as a surprise to some [img]/forums/images/icons/tongue.gif[/img] , it is not my mission in life to forever debate semantics, semiology or nomenclature on the Probability page of the twoplustwo.com website. Hence, and since my lowly contribution here is already taking too much bandwidth, this is my last post in this thread.

[BruceZ]

Some time ago I posted about what is known as "Simpson’s paradox". It is named as such in statistics books since this has become popular nomenclature, but they will also tell you that it is not a paradox at all, but simply a counterintuitive arithmetic fact.

Simpson's Paradox

There are many examples of misnomers in science and mathematics. Centrifugal force is not a real force, most random numbers are not really random, and there is no such thing as rms power no matter what your stereo specs say.

When mathematicians refer to something as a paradox which is not really a paradox, they are either doing so because of popular terminology, or because they do not really understand the problem under consideration. Mathematicians are not experts in all areas of mathematics, and not all mathematicians are competent statisticians.

As to the issue of whether all paradoxes will eventually be resolved when we finally know everything, mathematics has proven that this can never happen if we limit our knowledge to that which can be proven by mathematical logic. There will always be statements that can neither be proved nor disproved by logic, otherwise the theories in which those statements arise would become inconsistent. On the other hand, theories about the universe need not depend on the completeness of mathematical logic. For example, physics is not mathematics. Physics uses mathematics, it even creates its own mathematics, but theories of physical reality do not constitute a formal logical system in the mathematical sense. It is not of particular concern to a physicist if the mathematical theory which he uses is incomplete in the sense that there are statements which can neither be proved nor disproved, unless those statements come up in a physical context. For this reason, one might suppose that physicists may at least have a shot at eventually believing they understand everything about the known physical universe, but they will never convince mathematicians that they know everything.

[BruceZ]

I don't even consider the Monty Hall problem, with all information given, to be an example of a paradox in the everyday sense of the word. At least it is not for me, and not for anyone who thinks about the problem the right way. If you never switch, you will be wrong 2/3 of the time, so if you always switch you must be right 2/3 of the time. That is indisputable. This is the simplest way to think about the problem, and there is nothing counterintuitive about it. To say it doesn't matter if you switch would be to suggest that Monty opening a door suddenly makes you win at a higher rate when you do nothing differently, which would be truly counterintuitive, and for good reason since it would be clearly wrong. This notion can be immediately dismissed. People who are not used to thinking about conditional probability can confuse themselves by thinking about this problem other ways, but you don't even need conditional probability to analyze this. There are other problems where you do need conditional probability which are far more counterintuitive.

[Al Mirpuri]

The only factual inaccuracy, as opposed to a disagreement on the value of any specific point, I could find with your post was that you deny that the Monty Hall Paradox has anything to do with philosophy. This is not so. I first came across the Monty Hall Paradox in a philosophical dictionary whilst studying for a philosophy degree.

My post has generated a response that is admirable for its intellectual rigour. There is obviously much to discuss. As humans, I think the truth is beyond us; so let the debate rumble on.

For those of you who think that mathematics and science are rigorous disciplines that allow no room for disagreement or subjectivity (the 'I'm right, You're wrong' school) can I provide two examples. First, any equation concerning Pi attains an arbitrary certainty for Pi cannot be fully calculated and this spurious exactitude is given to Pi as there is a very human need for certainty. Secondly, positive co-ordinates are plotted up and right on graphs whilst negative co-ordinates are plotted down and left on graphs and this is because we are a right-handed species that believes heaven is above us whilst fearing left-handers and thinking hell is beneath us. In short, mathematics is value-laden (just as science is).

[BruceZ]

For those of you who think that mathematics and science are rigorous disciplines that allow no room for disagreement or subjectivity

I certainly think that the science of mathematics is rigorous, the physical sciences somewhat less so, and other sciences less so still. I certainly do NOT think that there isn't room for disagreement and subjectivity in any of these sciences. Subjectivity in mathematics arises from the creation of axioms which assign arbitrary truth or falsehood to statements that cannot be proven, and also from the making of arbitrary definitions. Different mathematicians may make different arbitrary choices of definitions and axioms based on their opinions of what should make mathematics most elegant, or most powerful, or they may do so simply to explore where a different set of axioms will lead. Once a particular choice of axioms and definitions are agreed upon however, there can be no valid dispute as to what has been proven and what has not been proven.

The truth of the Monty Hall problem is absolutely NOT "beyond us". It is completely known, quite trivial, and agreed upon by all competent mathematicians. The two different answers to the Monty Hall problem you gave are each only valid for a specific set of conditions which must be stated in the problem, and they are not simultaneously valid for any completely specified set of conditions. Failure to state a complete set of conditions is the only valid reason why there ever would be any debate among competent mathematicians. If there is not enough information specified to answer the question (and I believe this is almost always the case because my condition c is virtually always ignored) then different mathematicians could make their own separate assumptions to fill in the missing information. Then and only then would each of these answers have the same validity. Even then there would be no cause for a debate because once each mathematician states his assumptions, then all other competent mathematicians must agree that his answer follows from those assumptions. If there is not this agreement, then it means necessarily that some of the mathematicians are not competent, either because they analyzed the problem wrong, or because they failed to state their assumptions. If a philosophy book states otherwise, then the author of that book may be a competent philosopher, but he would be an incompetent mathematician. Incompetence does not justify declaring mathematical conclusions to be subjective.

See my posts below for the proper way this problem should be stated, and the one and only one correct solution given this problem statement. If after reading my problem statement and explanation you still believe that not switching is a valid solution, then there will be nothing more I can do to impart to you an understanding of this matter; however, I do have a game I want to play with you in which I will give you 3-2 odds on what you will believe to be an even money shot. You should be eager to play this game with me for as long as I like. I will also be eager to play this game with you until one of us is broke, because I will know with absolute certainty that you are a 2-1 underdog. Gambling is one arena where I welcome people to disagree with me.

Now for your two examples of subjectiveness in mathematics. The fact that pi is irrational does not mean that we can not state formulas involving pi which are completely exact and certain under the axioms of mathematics. Pi is a very precisely defined quantity. The fact that we graph things a certain way is an arbitrary convention, and does not produce subjective truths.

In the physical sciences, different scientists may hold different opinions as to what they believe to be true. It is never really possible to prove that a theory is true in the physical sciences. The best we can do is mount experimental evidence for a particular theory so that the theory becomes verified to a greater degree. It IS possible to DISprove a theory with a single experimental result however, so long as the experiment is reproducible by other scientists, and so long as there is no dispute about the accuracy of the experiment.

[sillyarms]

Check this out.
monty hall problem

[Al Mirpuri]

Since I replied to your post, I have gone back to the source of my original knowledge of what I termed the 'Monty Hall Paradox'. Having done so, I now hold that 'Monty Hall' is a problem and not a paradox and that correct reasoning leads to a switching of choices everytime.