// DEPENDENCE So, if this is the beginning, where do we go from here? Of course, we need to return to the very beginning of our story. The beginning was about the distinction between truth and assumption and how easy it is to become dualistic (the right way and the wrong way) in our approach to the truth. The subtleness in this section will highlight a very important concept, dependence, which is often overlooked, even by people who should know better. If you are like most mathematical modelers, you will be more than content to get your model to work. That is, you get an answer to a tricky question. It may take years of aggravating work and countless setbacks before you see any results, but alas, you see an answer! I would say that it is only human to form an attachment to your answer and for some it may take the form of a religious conviction. Religious convictions are hard to let go of and people can get quite ugly when confronted with evidence that their epiphany may be a mere illusion. Let s start the journey. The Sunday, September 9, 1990 issue of Parade magazine carried an interesting mathematical teaser in the Ask Marilyn section. The Ask Marilyn moniker is in reference to Marilyn vos Savant who is listed in The Guinness Book of World Records Hall of Fame for having an exceptionally high I.Q.. Here is the mathematical teaser, verbatim: The Question: Suppose you re on a game show, and you re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, do you want to pick door No. 2? Is it to your advantage to switch your choice? Craig F. Whitaker, Columbia, Md. Marilyn vos Savant s answer: Yes; you should switch. The first door has a one-third chance of winning, but the second door has a two-thirds chance. Here s a good way to visualize what happened. Suppose there are a million doors, and you pick door No. 1. Then the host, who knows what s behind the doors and will always avoid the ones with the prize, opens them all except door #777,777. You d switch to that door pretty fast, wouldn t you? Of course Ms. vos Savant doesn t have endless space to explain, but rather presents a terse explanation that offers what many accept to be the truth. I hope you re not convinced though and are willing to run a computer simulation. Here s mine:
PAGE 56 #include <iostream> // needed for cout #include <cstdlib> // needed for rand() and RAND_MAX using namespace std; const int LIMIT = 1000000; double uniform(void); // function U(*) prototype double uniform(void) return(static_cast<double>(rand()) / RAND_MAX); } int main (int) int door[4]; double probability; int contestant_choice; int host_choice; int i; int win = 0; for (i = 1 ; i <= LIMIT ; i++) // initialize doors door[1] = door[2] = door[3] = 0; // no prizes yet probability = uniform(); if (probability <= 1.0/3.0) // 0/3 <= p <= 1/3 door[1] = 1; // winning door if (probability <= 2.0/3.0) // 1/3 < p <= 2/3 door[2] = 1; // winning door // 2/3 < p <= 1 door[3] = 1; // winning door // contestant selects door probability = uniform(); if (probability <= 1.0/3.0) contestant_choice = 3; if (probability <= 2.0/3.0) contestant_choice = 2; contestant_choice = 1; probability = uniform(); switch (contestant_choice) // decide what door the host will open case 1: if ( door[1] == 1) if (probability <= 1.0/2.0) host_choice = 2; host_choice = 3; if (door[2] == 1) host_choice = 3; host_choice = 2; case 2: if ( door[2] == 1) if (probability <= 1.0/2.0) host_choice = 1; host_choice = 3; if (door[1] == 1) host_choice = 3; host_choice = 1; case 3: if ( door[3] == 1) if (probability <= 1.0/2.0) host_choice = 1;
PAGE 57 } host_choice = 2; if (door[2] == 1) host_choice = 1; host_choice = 2; default: cout << "ERROR!\n"; switch (host_choice) // decide what door the contestant will switch to case 1: if (contestant_choice == 2) contestant_choice = 3; contestant_choice = 2; case 2: if (contestant_choice == 1) contestant_choice = 3; contestant_choice = 1; case 3: if (contestant_choice == 2) contestant_choice = 1; contestant_choice = 2; default: cout << "ERROR!\n"; } // win? if (door[contestant_choice] == 1) win++; } // for cout << "If you switch: " << win << " wins out of " << LIMIT << " trials.\n"; } // main The Output: If you switch: 666645 wins out of 1000000 trials. Wow, Ms. von Savant was right! After all, a simulation on a trusted machine produced almost exactly two-thirds as stated. You could even use Bayesian analysis to get this result, but that s not the point anyway. Briefly, Bayesian analysis looks at prior events to figure out the probability of a future event and we ll talk more about that later. The story doesn t stop here though and will probably rage on for years to come. Just do a search on Google (http://www.google.com/) and you will see a plethora of web sites on this one little tidbit. In fact, this one little tidbit in Parade got promoted to the front page of The New York Times, which is perhaps where some of the world s most important truths have come to light for the general reader.
PAGE 58 The detail is in the lower right hand quarter: Behind Monty Hall s Doors: Puzzle, Debate and Answer? The article was written by John Tierney for the July 21, 1991 issue of the New York Times. PHOTOCOPY OF THE FRONT PAGE AND ARTICLE TEXT IS USED WITOUT PERMISSION OF THE NEW YORK TIMES OR MR. TIERNEY. I BELIEVE THIS IS CONSIDERED FAIR USE UNDER US COPYRIGHT LAW AS I UDERSTAND IT. The article s front-page position and lengthy stance certainly forebodes possible enlightenment and it s worth more than just a casual glance. Here s the full text of the article in all its glory: Behind Monte Hall s Doors: Puzzle, Debate and Answers? By JOHN TIERNEY Special to The New York Times Beverly Hills, Calif., July 20 Perhaps it was only an illusion, but for a moment here it seemed that an end might be in sight to the debate raging among mathematicians, readers of Parade magazine and fans of the television game show Let s Make a Deal. They began arguing last September after Marilyn vos Savant published a puzzle in Parade. As readers of her Ask Marilyn column are reminded each week, Ms. vos Savant is listed in the Guinness Book of World Records Hall of Fame for the Highest I.Q., but that credential did not impress the public when she answered this question from a reader: Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, Do you want to pick door No. 2? Is it to your advantage to take the switch? Since she gave her answer, Ms. vos Savant estimates she has received 10,000 letters, the great majority disagreeing with her. The most vehement criticism has come from mathematicians and scientists, who have alternated between gloating at her ( You are the goat! ) and lamenting the nation s innumeracy. Her answer that the contestant should switch doors has been debated in the halls of the Central Intelligence Agency and the barracks of fighter pilots in the Persian Gulf. It has been analyzed by mathematicians at the Massachusetts Institute of Technology and
PAGE 59 computer programmers at Los Alamos National Laboratory in New Mexico. It has been tested in classes from second grade to graduate level at more than a 1,000 schools across the country. But it was not until Thursday afternoon that a truly realistic simulation of the problem was conducted. The experiment took place at the Beverly Hills home of Monte Hall, the host of 4,500 programs of Let s Make a Deal from 1963 to 1990. In his dining room Mr. Hall put three miniature cardboard doors on a table and represented the car with an ignition key. The goats were played by a package of raisins and a roll of Life Savers. After Mr. Hall allowed this contestant 30 independent attempts to win the car, two conclusions seemed clear: Ms. vos Savant s vitriolic critics, including the mathematics professors, are dead wrong. But Ms. vos Savant is not entirely correct either, because there is a small flaw in her wording of the problem that was detected not only by Mr. Hall but also by some of the experts. Despite her impressive analysis and 228-point I.Q. she was not as quick as Mr. Hall in understanding the psychological dimensions of the problem. So Easy to Blunder A few mathematicians were familiar with the puzzle long before Ms. vos Savant s column. They called it the Monte Hall Problem the title of the analysis in the journal American Statistician in 1976 or sometimes Monty s Dilemma or the Monty Hall Paradox. An earlier version, the Three Prisoners Problem, was analyzed in 1959 by Martin Gardner in the journal Scientific American. He called it a wonderfully confusing little problem and presciently noted that in no other branch of mathematics is it so easy for experts to blunder as in probability theory. The experts responded in force to Ms. vos Savant s column. Of the critical letters she received, close to 1,000 carried signatures with Ph.D. s and many were letterheads of mathematics and science departments. Our math department had a good, self-righteous laugh at your expense. wrote Mary Jane Still, a professor at Palm Beach Junior College. Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va., expressed the prevailing view that there was no reason to switch doors. You blew it! he wrote. Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice neither of which has any reason to be more likely to 1 2. As a professional mathematician, I m very concerned with the general public s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful. The criticism has continued despite several more columns by Ms. vos Savant defending her choice. You are utterly incorrect, insisted E. Ray Bobo, a professor of mathematics at Georgetown University. How many irate mathematicians are needed to get you to change your mind? The Henry James Treatment Mr. Hall said he was not surprised at the experts insistence that the probability was 1 out of 2. That s the same assumption contestant s would make on the show after I showed them there was nothing behind one door, he said. They d think the odds on the door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered. By opening that door we were applying pressure. We called it the Henry James treatment. It was The Turn of the Screw. Mr. Hall said he realized the contestants were wrong, because the odds on Door 1 were still only 1 in 3 even after he opened another door. Since the only other place the car could be was behind Door 2, the odds on that door must now be 2 in 3. Sitting at the dining room table, Mr. Hall quickly conducted 10 rounds of the game as the contestant tried the non-switching strategy. The result was four cards and six goats. Then for the next 10 rounds the contestant tried switching doors, and there was a dramatic improvement: eight cars and two goats. A pattern was emerging. So her answer s right: you should switch, Mr. Hall said, reaching the same conclusion as the tens of thousands of students who conducted similar experiments at Ms. vos Sanant s suggestion. That conclusion was also reached eventually by many of her critics in academia, although most did not bother to write letters of retraction. Dr. Sach s, whose letter was published in her column, was one of the few with the grace to concede his mistake. I wrote her another letter, Dr. Sachs said last week, telling her that after removing my foot from my mouth I m now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It s been an intense professional embarrassment. Manipulating a Choice Actually, many of Dr. Sachs s professional colleagues are sympathetic. Persi Diaconis, a former professional magician who is now a Harvard University professor specializing in probability and statistics, said there was no disgrace in getting this one wrong. I can t remember what my first reaction to it was, he said, because I ve known about it for so many years. I m one of the many people who have written papers about it.
PAGE 60 But I do know that my first reaction has been wrong time after time on similar problems. Our brains are just not wired to do probability problems very well, so I m not surprised there were mistakes. After the 20 trials at the dining room table, the problem also captured Mr. Hall s imagination. He picked up a copy of Ms. vos Savant s original column, read it carefully, saw a loophole and then made a suggested more trials. On the first, the contestant picked Door 1. That s too bad, Mr. Hall said opening Door 1. You ve won a goat. But you didn t open another door yet or give me a chance to switch. Where does it say that I have to let you switch every time? I m the master of the show. Here, try it again. On the second trial, the contestant again picked Door 1. Mr Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills. You re sure you want Door No. 2? he asked. Before I show you what s behind that door, I will give you $3,000 in cash not to switch to it. I ll switch to it. Three thousand dollars, Mr. Hall repeated, shifting into his famous cadence. Cash. Cash money. It could be a car, but it could be a goat. Four thousand. I ll try the door. Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousands. Let s open the door. You just ended up with a goat, he said, opening the door. The Problem With the Problem Mr. Hall continues: Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind door 1. That s the kind of thing I can do when I m in control of the game. You may think you have probability going for you when you follow the answer in her column, but there s the psychological factor to consider. He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant s advice and stick with the original door. Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and usually did not offer it. And although Mr. Hall might have been violating the spirit of Ms. vos Savant s problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant s wording of the problem with the versions they had analyzed in their articles. The problem is not well-formed, Mr. Gardner said, unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it s to his advantage to let the player switch, and the probability of being right by switching could be as low as zero. Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch. Ms. vos Savant acknowledged that the ambiguity did exist in her original statement. She said it was a minor assumption that should have been made obvious by her subsequent analysis, and that it did not excuse her professional critics. I wouldn t have minded if they had raised that objection, she said Friday, because it would mean they really understood the problem. But they never got beyond their first mistaken impression. That s what dismayed me, Still, because of the ambiguity in the wording, it is impossible to solve the problem as stated through mathematical reason. The strict argument, Dr. Diaconis said, would be that the question cannot be answered without knowing the motivation of the host. Which means, of course, that the only person who can answer this version of the Monty Hall Problem is Monty Hall himself. Here is what should be the last word on the subject: If the host is required to open the door all the time and offer you a switch, then you should take the switch, he said. But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood. My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home. What is the point though? The main point is that events may be unrelated to the prior events, however we shouldn t expect this to be the case always. A fair coin, for example, is as likely to come up heads as it is tails regardless of what has happened in
PAGE 61 the coin s prior history. The coin doesn t think and each new toss is independent of the coin s past and certainly has no effect on the coin s future! Nevertheless, probabilities on a human level are changing depending on the events as they unfold. A human thinks about the consequence, the contestant wants to win and Mr. Hall may, or may not want the contestant to win. Each person in this game needs to attach a subjective probability to his or her individual decisions and those probabilities may appear to change erratically especially on a highly entertaining game show where time restraints and audience perceptions can easily influence both Mr. Hall and the contestant. Mr. Hall s skills as a host are largely dependent on his audiences perceptions and keep in mind that he must have been pretty damn good to last twenty-seven years. We need to think about how the real world works when modeling a simulation involving human participants. Dependence can often lead to an unexpected difference between what s predicted and what actually happens. Unfortunately, you will often hear mathematical modelers making the independence assumption; usually they ll just say that the model depends on independent and identically distributed random variables and it is said so often that it has its own abbreviation i.i.d.. All the computer simulations in this text have been assumed to be from an i.i.d. random variable and that will not change in this text. The switching strategy has a probability of winning two-thirds of the time only if Mr. Hall acts like a machine. Mr. Hall is anything but mechanical: he s a consummate and a highly intelligent salesman. Furthermore, your computer simulation may make you think the probability is fixed but the reality, due to dependence, can actually be much different. Further analysis of Mr. Hall s past behavior may make us better able to judge each new situation, but the situation will remain dynamic and hard to predict. By dependence, I mean that the probability of what happens next is somehow dependent on some prior event or events. Humans are not machines and when we try to model their behavior using computer simulation, we may be disappointed in how poorly we predict the actual event especially if we think their behavior is independent of past events. My untested hypothesis about Monte Hall is that he was probably biased in his dealings with contestants. No one can be 100% fair and for whatever reason Mr. Hall may have tried to persuade contestants into losing, or for that matter, winning. Why?
PAGE 62 Well, maybe it was something psychologically subtle that goes beyond our sense of detection. Even in the article, you can see where Mr. Hall shifts gears to show he is in control by not giving a choice, or even forcing a bad choice. I am going to call this a strategic uncertainty principle, simply stated: The more precisely you know the strategy of an opponent game player, the less likely their strategy will remain the same. In a sense, predictability destroys the fun. For an interesting application of this uncertainty principle, I would suggest looking at the Black-Scholes option pricing model developed by two Nobel laureates as was used by Long-Term Capital, an investment firm. Essentially Long-Term Capital generated limitless wealth in the financial markets, all without work, but then the market decided it didn t like their ways and punished them with limitless losses. Essentially humans are not computers and they will change and adapt to profit from computer like behavior. Especially true if they are allowed to do so.