Two of the letters I received warrant further discussion. I would like to quote from them, and respond to them as well as I may.
keeps this site and its author alive.
From: "John A. Major" (e-mail address withheld as a courtesy)
Subject: vos Savant puzzle
Dear M. J. Young
Thank you for posting what I consider to be the best solution to the "three doors" puzzle I have seen to date. I recall a FAQ in one of the alt.sci newsgroups which claimed to be the "last word" but clearly was not. In searching for it through Alta Vista, I came upon you page.
I would like to add three points.
1) Ex post, there is no "probability" involved. One either has the right door (probability one) or not (probability zero), and which is the case will become known after all doors are open. The typical interpretation imposed by puzzle solvers is of the "frequentist" type and refers to the long-run relative frequencies of outcomes in many "plays" of the game. And that is where the strategy of the game show host becomes an issue -- what are the possible outcomes we should be mentally tabulating?
2) It is possible to consider a malicious host that only offers you the chance to switch if you already have the right door (you did mention that the host may not be required to offer you the right door!). If that is the host's strategy, then an offer to switch provides a probability of zero that the offered door is the right one.
3) Given the span of strategies available to the host (and they can be "mixed," e.g., by rolling dice to decide what strategy to follow), a two-person zero-sum game theory formulation leads to the recognition that holding on to the first door is a "saddle point." The player can guarantee that no matter what the host does, he has at least a 1/3 chance of having the right door. The host's optimal strategy, on the other hand, is to offer a second door only if the player has the right one; in this way he can guarantee that the player's chance is at most 1/3. Therefore, if the question is "Should the player switch?" then the answer is "No."
John A. Major, ASA....
From: "Phil Schipsi (8-276-7657)" (e-mail address withheld as a courtesy)
Subject: Ask Marilyn (Monte Hall Problem)
I must take exception to your analysis of the Q/A on this problem. You have reconstructed the problem beyond all recognition. In its original form it goes like this: You are to select one of three doors, behind one of which there is a Cadillac. Behind the other two are goats. After you have selected a door, the host reveals the content behind a door of his selection. You are then permitted to swap your door. The host knows which door hides the prize. In this scenario, the (only) one confronting Marilyn, (statistically) one should ALWAYS trade. PERIOD! There are no "ASSUMPTIONS" to deal with! The problem, as you have stated it, does not and would not, have made any sense. Nor was that the way it was printed in PARADE. Explanation: The host is constrained to show a door containing a goat. If you have selected the prize-door, then his presentation of a goat is not different from his presenting a goat when you have selected one. The latter will have occurred at a two to one ratio to the former. If you have selected a goat, he must still reveal a goat. To do otherwise would violate the rules of any game that makes sense. Imagine, if you will, that he reveals the Cadillac, and asks you whether you want to switch.
If you don't have a copy of the original question - and I believe that you don't - then the issue becomes moot. But if you do, please re-read it. There is no conditional assumption, that would affect the outcome. If you believe that there is, then you have not read the same problem that I have read, and I DID READ IT.
Respectfully, Phil Schipsi
P.S. She has answered many questions incorrectly. This wasn't one of those.
I would begin by saying that Phil is right at least on this point: I never saw the original question. It was first posed to me by telephone, in a rather off-the-cuff manner, described rather than read. My conclusion immediately was as stated in my page--and the problem which I was given also was as stated in my page. I was then sent a cutting of Ms. Savant's column; this was not the original statement of the problem, but a restatement in a subsequent column in which her position was bolstered by a group who had programmed a computer to answer the same question. The question as stated in that column was not different from that which appears on the page in which I discuss it; yet I will admit readily that it may have been altered from the original. On the other hand, I still have not seen the original column, and any statement of the contents of that column which does not quote it directly and completely must be regarded hearsay, of no greater validity than the statement of the problem which I first heard, and intrinsically less reliable than Ms. Savant's own restatement thereof in the follow-up column. Still, the facts of the problem are vital to its conclusion, so a consideration of the suggested alternate statement of the problem is warranted. But I would prefer to address the letters in the order in which I received them--and not only for that reason, but also because it will make more sense in the entirety.
Mr. Major suggests that "there is no 'probability' involved". He refers to the "frequentist" type of probability. I understand what he is saying, but I think there is a distinction here. Indeed, the type of probability involved here is different from that with dice. Until the dice are rolled, the number which will be rolled is unknown and strictly a matter of probabilities based on random events. However, the doors represent the same type of probabilities problem which occurs in card games: each of the cards is what it is, and therefore the next card in the deck is certain, not probable; yet to me not knowing the factual identity of that card, its identity is a probability. If I hold three aces and one king in a hand of poker, five-card draw, when I am called upon to draw, I may discard the remaining card in a hope to draw either the remaining ace or one of the three kings. Since to me exactly five cards are known--those in my hand--there are forty-seven cards unknown to me, and the top card could, from my perspective, be any one of them. Four of those forty-seven will benefit me; so my chance of drawing a card which benefits me is four in forty seven, almost one in twelve. In fact, that card is already fixed, in that it is what it is. In fact, there are five other cards that it cannot be--the five held by my opponent. In some sense, it cannot be any of the other forty cards beneath it, either, because it is what it is. But I do not know what card is there, or in my opponent's hand, or anywhere else. Thus relative to my knowledge of the situation the identity of the card is a matter of probability. Note that it would be exactly the same if the dice in a craps game were rolled but obscured from my view, perhaps hidden under the dice cup. As long as I do not know what the roll is, there are six chances in thirty-six that it is a seven, and two chances in thirty-six that it is an eleven. That the roll has been made does not change what it is likely to be; it only determines what it will be. The situation is the same with the three doors: as long as I do not know what is behind any door, the probability that I have the correct door is one in three.
However, Mr. Major also suggests that there is a difference between the probability of a specific result in one play of the game and the probability of a result in repeated plays of the game. I have no formal education in probabilities--I've studied them primarily in connection with games--and so am not certain whether there might be some validity to his distinction of the "frequentist" probability. However, in my experience the one is merely a reflection of the other. That is, when I roll a six-sided die, there are six possible outcomes, each equally likely. If I must roll any one specific number to succeed, I have one chance in six of success, as there is one outcome which is success and five which are failure, each equally likely to occur. And so if I am to play the game only once, my chance of winning is one in six. Yet if I roll that same die six thousand times, roughly one thousand of those rolls will be the number I require--one success in six rolls. The "frequentist" probability exists as an echo of the single case probability; whenever the chance of success in a single case is calculable, it logically plays out that in repeated cases the results will reflect that probability.
(In the case of rolling the six-sided die, it is unlikely that each number will occur once in six rolls, since the probability on each roll of rolling a number not rolled on one of the previous rolls decreases--6/6 on the first roll, 5/6 on the second, 4/6 on the third, until on the sixth roll--assuming all previous rolls were unique results--the chance is 1/6--16% of 33% of 50% of 67% of 83% of 100%, about one and a half percent of rolling each number exactly once on six rolls.)
Mr. Major also suggests the possibility of the "malicious host" who only offers you the chance to switch to another door if you have in fact chosen the correct door. Observing that my analysis allowed that the host might eliminate the right door, he suggests that the host might not be compelled to offer any door, and so would gain by offering a losing door when the player has selected a winner, but would have no reason to offer a door when the player has selected a loser already. In regard to this, it was my reading of the problem that the host was required to offer a door. However, given that I did not have the original problem, let us assume that I am mistaken, and the host's offer of an alternate door is an option. Mr. Major suggests that the winning strategy for the host is to end the game when you choose a wrong door first, but to offer you a wrong door when you choose a right door. I agree that that strategy will always work the first time. However, if we assume that the game will be played repeatedly, and that the player will have knowledge of the outcome of previous games, it becomes necessary for the host to give away the winning door at least sometimes. The casino owners know that big winners are good for business, because they encourage the bigger losers; and although the odds are better to be killed by street crime, people buy lottery tickets who would scream quite loudly if the state charged them so much in taxes. The winning strategy in the long haul requires that the host at least create the illusion that you have a chance to win by changing doors, and therefore he must sometimes offer the winning door to at least some players. Thus even a malicious host must give away the winning door sometimes. However, Mr. Major is on the right track here--the host need only create the illusion that you can win the right door by trading, and so he need not offer it so often as even odds. Given the possibility suggested by Mr. Major, that the host might know which door had the prize and wish for the player to lose, the odds prefer staying with the original door. This is because the chance of selecting the right door on the first guess is one in three, but the chance that the host will offer you the right door is less than that. Although it is a rather subjective estimation, it is reasonable to suggest that the host can only win if he offers you the winning door on less than half of the two-thirds of the times when you don't pick it on the first pick, thus less than one third of the times. Given the possibility that the host could wish to mislead you into selecting the wrong door and have the means to do so, your odds are best if you stay with the first door you select.
And since under the scenario described, we aren't told whether the host knows which door is which or what his motives are, Mr. Major is also correct that your odds are better if you do not change to the other door when it is offered.
However, Mr. Schipsi has a very different take on the subject. He claims that the problem as stated in my page is not the problem answered by Ms. Savant. According to him, there are several points which are different in the original problem which are critical to the analysis. First, he says that we know what the prizes are (I had never heard these prizes mentioned before, so I know I've never seen the statement of the problem he is citing). Second, although in my understanding of the problem the host would eliminate the third door from consideration merely by offering you the second, Mr. Schipsi claims that the host is required to reveal what is behind the eliminated door before offering you the option to switch to the remaining door. Intrinsic in this is the requirement which I understood (but which Mr. Major denied), that the host must offer you an alternate door regardless of what you have already selected. Also, we are given that the host has knowledge of what is behind each door. In short, all of the assumptions which I say were made improperly Mr. Schipsi says were part of the original problem.
It is certainly possible that Mr. Schipsi is correct about the original statement of the problem, which I never saw. The restatement of the problem I read in the later column did not hint to these facts--but that is very like entering a conversation in the middle, and I may well have missed those critical facts. What is of greater concern to me is that Mr. Major, who suggests that he has been following the question from the beginning, is patently unaware of at least some of those claimed facts. For example, if the host is required to offer an alternate door, then Mr. Major's suggested strategy of offering a door only if the player chooses the right door first is clearly impossible; further, if the host must reveal what is behind the door eliminated and the nature of the prizes are known in advance, then it is not possible to offer a wrong door if the player has selected a wrong door, since that would require showing the character the prize behind the right door, and then asking if he would wish to trade one clearly wrong door for the other! Even were we to suggest that Mr. Major is a complete idiot (not at all warranted in view of his intelligent observations on the problem), it is difficult to imagine that he could make those suggestions had he read the problem stated by Mr. Schipsi. We are forced to choose between limited options. Possibly one or both of our two correspondents did not read the original problem; possibly one or both of them forgot the original problem, or is remembering a version of the problem which mixes aspects of the original problem with apocryphal elements added over time to help explain the problem, or which edits sections of the problem incorrectly deemed superfluous.
Whatever the cause, it is clear that the problem addressed by Mr. Major is entirely different in its critical facts from the one addressed by Mr. Schipsi. And the two distinct problems have opposite solutions. For Mr. Schipsi and Mr. Major agree that the player's chance of selecting the correct door originally is one in three. But if the player chooses the wrong door, under Mr. Major's statement of the problem, the probability that the host will offer the right door is low--not as high as fifty percent--while under Mr. Schipsi's statement of the problem, the probability that the host will offer the correct door when the player has not chosen it is one hundred percent.
Thus, under Mr. Schipsi's statement of the problem (which he is quite adamant is correct), Ms. Savant's solution is correct, and I must apologize for...for solving the wrong problem.
Thus I offer a conditional apology: if the problem I read was not the problem as originally stated, I apologize for suggesting that the solution to the problem I read was the solution to the problem I did not read. At the same time, the solution I gave is the solution to the restated problem as I received it.
I again thank Mr. Major and Mr. Schipsi for their insightful letters on the matter.
Return to my original discussion of the problem.