The Monty Hall Problem is not a Probability Puzzle (its a challenge in mathematical modelling)

The Mon t y Hall Problem is not a Probabilit y Puzzle ∗ (It’s a c hallenge in mathematical mo delling) Ric hard D. Gill † 12 No v ember, 2010 Abstract Supp ose you’r e on a game show, and you’r e given the choic e of thr e e do ors: Behind one do or is a c ar; b ehind the others, go ats. Y ou pick a do or, say No. 1, and the host, who knows what’s b ehind the do ors, op ens another do or, say No. 3, which has a go at. He then says to you, “Do you want to pick do or No. 2?” Is it to your advantage to switch your choic e? The answ er is “yes” but the literature offers many reasons why this is the correct answer. The present pap er argues that the most common reasoning found in introductory statistics texts, dep ending on making a num b er of “obvious” or “natural” assumptions and then computing a conditional probability , is a classical example of solution driv en science. The b est reason to switch is to b e found in von Neu- mann’s minimax theorem from game theory , rather than in Ba yes’ theorem. 1 In tro duction In the ab ov e abstract to this pap er, I reproduced The Mon t y Hall Problem, as it was defined by Marilyn v os Sa v ant in her “Ask Marilyn” column in Par ade magazine (p. 16, 9 September 1990). Marilyn’s solution to the prob- lem posed to her by a corresp onden t Craig Whitaker sparked a con trov ersy ∗ v.4. Revision of v.3 , to appear in Statistic a Ne erlandic a † Mathematical Institute, Univ ersit y Leiden; during 2010–11 the author is Distin- guished Lorentz F ello w at the Netherlands Institute of Adv anced Study , W assenaar. http://www.math.leidenuniv.nl/ ∼ gill 1 whic h brought the Mont y Hall Problem to the attention of the whole w orld. Though MHP probably originated in a pair of short letters to the editor in The Americ an Statistician b y biostatistician Steve Selvin (1975a,b), from 1990 on it w as public prop erty , and has spark ed research and contro versy in mathematical economics and game theory , quan tum information theory , philosoph y , psyc hology , ethology , and other fields, as well as having b ecome a fixed p oin t in the teaching of elementary statistics and probabilit y . This has resulted in an enormous literature on MHP . Here I would lik e to dra w attention to the splendid b o ok by Jason Rosenhouse (2009), which has a h uge reference list and which discusses the pre-history and the p ost-history of vos Sav an t’s problem as w ell as man y v ariants. My other fa vourite is Rosen thal (2008), one of the few pap ers where a genuine attempt is made to argue to the layman wh y MHP has to b e solved with conditional probability . Aside from these t w o references, the English language wikip edia page on the Mon ty Hall Problem, as well as its discussion page, is a rich though every- c hanging resource. Muc h that I write here was learnt from the man y editors of those pages, b oth allies and enemies in the never ending edit-w ars whic h plague it. The battle among wikip edia editors could b e describ ed as a battle b et w een in tuitionists v ersus formalists, or to use other words, b et ween simplists v ersus conditionalists. The main question which is endlessly discussed is whether simple argumen ts for switching, which typically show that the unc onditional probabilit y that the switching gets the car is 2/3, may b e considered rigor- ous and complete solutions of MHP . The opp osing view is that vos Sav an t’s question is only prop erly answered b y study of “the” c onditional probabil- it y that the switching gets the car, given the initial choice of do or by the pla yer and do or op ened b y the host. This more sophisticated approach re- quires making more assumptions, and that leads to the question whether those supplemen tary conditions are implicitly implied by v os Sav an t’s words. What particularly interests me, how ev er, is that the conditionalists tak e on a dogmatic stance: their p oin t of view is put forward as a moral imp erativ e. This leads to an impasse, and the clouds of dust thro wn up by what seems a religious w ar conceal what seem to me to b e muc h more interesting, though more subtle, questions. My p ersonal opinion on the wikipedia-MHP-wars is that they are fights ab out the wrong question. Craig Whitak er, through the voice of Marilyn v os Sa v ant, asks for an action, not a probability . I think that game theory giv es a more suitable framework in which to represen t our ignorance of the mec hanics of the set-up (where the car is hidden) and of the mec hanics of the host’s c hoice, than sub jectivist probability . Therefore, though Rosenhouse’s b o ok is a wonderful resource, I strongly 2 disagree with him, as well as with many other authors, on what the essential Mon ty Hall problem is (and that is the main point of this pap er). Deciding unilaterally (Rosenhouse, 2009) that a certain formulation is c anonic al is asking for schism. Calling a sp ecific version original (Rosen thal, 2008) is asking for contradiction. Rosenthal states without any argument at all that additional assumptions are implicitly contained in v os Sav an t’s formulation. Selvin (1975a) did state all those assumptions explicitly but strangely enough did not use all of them. His second pap er Sevin (1975b) ga v e a new solution using all his original assumptions but the author do es not seem to notice the difference. At the same time, he quotes with approv al a simplist solution of Mon ty Hall himself, who sees randomness in the c hoice of the pla yer rather than in the actions of the team who prepare the sho w in adv ance, and the quiz-master himself. V os Sav ant did not use the full set of assumptions whic h others find implicit in her question. Her relatively simple explanation of why one should switch seems to satisfy every one except for the writers of elementary texts in statistics and probability . I ha v e the impression that w ords like original, canonical, standard, complete are all used to hide the paucit y of argument of the writer who needs to make that extra assumption in order to b e able to apply the to ol whic h they are particularly fond of, conditional probabilit y . One of the most widely cited but p ossibly least w ell read pap ers in MHP studies is Morgan et al. (1991a), published together with a discussion b y Sey- mann (1991) and a rejoinder Morgan et al. (1991b). Morgan et al. (1991a) firmly rebuk e vos Sav an t for not solving Whitaker’s problem as they con- sider should b e done, namely b y conditional probabilit y . They use only the assumption that all do ors are initially equally likely to hide the car; this as- sumption is hidden within their calculations. The paper w as written during the p eak of public interest and heated emotions ab out MHP whic h arose from vos Sav an t’s column. It actually contains an unfortunate error which w as only noticed 20 years later by wikip edia editors Hogbin and Nijdam (2010): if the play er puts a non-informative and hence symmetric Bay esian prior on the host’s bias in op ening a do or when he has a c hoice, it will b e equally lik ely (for the play er) that the host will op en either do or when he has the opp ortunit y . Morgan et al. (2010) ac kno wledge the error and also repro- duce part of Craig Whitak er’s original letter to Marilyn vos Sav an t whose w ording is even more ambiguous than vos Sav an t’s. Rosenhouse (2009), Rosenthal (2005, 2008), Morgan et al. (1991a,b, 2010), and Selvin (1975b) (but not Selvin, 1975a) solv e MHP using elementary c on- ditional probability . In order to do so they are obliged to add mathematical assumptions to vos Sa v ant’s words, without whic h the conditional probabil- it y they are after is not determined. Actually , and I think tellingly , almost 3 no author giv es an y argument at all wh y we must solv e v os Sav ant’s question b y computing a conditional probability that the other do or hides the car, conditional on which do or was first c hosen b y the pla yer and whic h op ened b y the host. F or whatever reasons, it has b ecome conv entional in the elemen tary statis- tics literature, where MHP features as an exercise in the chapter on Bay es’ theorem in discrete probability , to tak e it for granted that the car is initially hidden “at random”, and the host’s choice, if he is forced to make one, is “at random” to o. Morgan et al. (1991a) are notable in only making the first assumption. Man y writers also ha v e the play er’s initial choice “at random” to o. “A t random” is a co de phrase for what I would prefer to call c ompletely at random. The student is apparen tly supp osed to mak e these assumptions b y default, though sometimes they are listed without motiv ation as if they are alw ays the right assumptions to make. In my opinion, this approach to MHP is an example of solution driven scienc e , and hence an example of bad practise in mathematical mo delling. T aking for granted that unsp ecified probability distributions must b e uniform or normal, dep ending on context, is the cause of such disasters as the mis- carriage of justice concerning the Dutch nurse Lucia de Berk, or the doping case of the German sk ater Claudia Pec hstein. Of course, MHP do es indeed pro vide a nice exercise in conditional probabilit y , provided one is willing to fill in gaps without whic h conditional probability does not help you answer the question whether y ou should stay or switc h. Morgan et al. (1991a)’s orig- inal contribution is to notice the minimal condition under which conditional probabilit y do es giv e an unequivocal solution. Precisely b ecause of all these issues, MHP presents a b eautiful playground for learning the art of mathematical mo delling. F or me, MHP is the problem of ho w to build a bridge from vos Sa v ant’s w ords to a mathematical problem, solv e that problem, and translate the solution back to the real world. If we use probability as a to ol in this enterprise, we are going to hav e to motiv ate probabilistic assumptions. W e must also interpr et probabilistic c onclusions . Lik e it or not, the in terpretation of probabilit y plays a role, going b oth directions. Real world problems are often brought to a statistician because the person with the question, for some reason or other, thinks the statistician m ust b e able to help them. The clien t has often already left out some complicating factors, or made some simplifications, which he thinks that the statistician do esn’t need to kno w. The first job of the consulting statistician is to find out what the real question is with which the clien t is struggling, which ma y often b e v ery differen t from the imaginary statistical problem that the clien t thinks he has. The first job of the statistical consultant is to undo the pre-pro cessing 4 of the question whic h his clien t has done for him. In mathematical mo del building we must b e careful to distinguish the parts of the problem statement whic h are truly determined by the bac kground real world problem, and parts whic h represent hidden assumptions of the clien t who thinks he needs to enlist the statistician’s aid and therefore has already pre-pro cessed the original question so as to fit in the client’s picture of what a statistician can do. The result of a statistical consultation might often b e that the original question p osed by the client is reform ulated, and the clien t go es home, happier, with a v aluable answer to a more meaningful question than the one he brought to the statistician. Maybe this is the real message whic h the Mont y Hall Problem should b e telling us? What if vos Sa v ant’s op ening words had b een “ Supp ose you’r e going to b e on a game show tonight. If you make it to the last r ound, you’l l b e given the choic e of thr e e do ors... ”? 2 The mathematical facts In this section, I present some elemen tary mathematical facts, firstly from probabilit y theory , secondly from game theory . The results are formulated within a mathematical framew ork which do es not make any assumptions restricting the scop e of the presen t discussion. Mo delling all the v arious do or c hoices as random v ariables do es not exclude the case that they are fixed. It also leav es the question completely op en how w e think of probabilit y: in a frequen tist or in a Bay esian sense. I imp ose only the “structural” conditions on the sequence of choices, or mov es, which are universally agreed to b e implied b y vos Sav an t’s story . 2.1 What probabilit y theory tells us I distinguish four consecutiv e actions: 1. Host: hiding the car b efore the show b ehind one of three do ors, Car 2. Play er: choosing a do or, P1 3. Host: rev ealing a goat by op ening a different do or, Goat 4. Play er: switching or staying, final c hoice do or P2 The do ors are conv en tionally labelled “1”, “2” and “3”, and we can rep- resen t the do or num b ers sp ecified by the four actions with random v ariables Ca r, P1, Goat, P2 . Since t w o do ors hide goats and one hides a car and the 5 host knows the lo cation of the car, he can and will open a door differen t to that chosen by the play er and rev ealing a goat. I allow b oth the lo cation of the car Ca r and the initial choice of the pla yer P1 to b e random, and as- sume them to b e statistically indep endent. F rom different mo delling points of view, we might wan t to take either of these tw o v ariables to b e fixed; the indep endence assumption is then of course harmless. Given the lo cation of the car and the do or c hosen b y the play er, the host op ens a different do or Goat revealing a goat, according to a probabilit y distribution ov er the tw o do ors av ailable to him when he has a c hoice (which includes the case that he follo ws some fixed selection rule). Then the play er makes his c hoice P2 , deterministically or according to a probabilit y distribution if he lik es, but in either case only dep ending on his initial choice and the do or op ened by the host. Finally we can see whether he go es home with a car or a goat b y op ening his do or and revealing his Prize . The probabilistic structure of the four actions together with the final result Prize (whether the play er go es home with a car or a goat) can b e represen ted in the graphical mo del or Bay es net shown in Figure 1. This diagram (dra wn using the gRain pack age in the statistical pack age R) was in- spired by Burns and Wieth (2004) who p erformed psyc hological exp erimen ts to test their h yp othesis that p eople fail MHP b ecause of their inability to in ternalise the c ol lider principle : conditional on a consequence, formerly in- dep enden t causes b ecome correlated. In this case, the initially statistically indep enden t initial mov es Ca r and P1 of host and pla yer are conditionally dep endent of one another given the do or Goat op ened b y the host. I no w write down three simple prop ositions, eac h making in turn a stronger mathematical assumption, and eac h in turn giving a b etter reason for switc h- ing. Prop osition 1. If the player’s initial choic e has pr ob ability 1 / 3 to hit the c ar, then always switching gives the player the c ar with (unc onditional) pr ob ability 2 / 3 (Mon ty Hall, as rep orted by Selvin, 1975b) . Prop osition 2. If initial ly al l do ors ar e e qual ly likely to hide the c ar, then, given the do or initial ly chosen and the do or op ene d, switching gives the player the c ar with c onditional pr ob ability at le ast 1 / 2 (Morgan et al., 1991a) . Conse quently, not only do es “always switching” give the player the c ar with unc onditional pr ob ability 2 / 3 , but no other str ate gy gives a higher suc c ess chanc e. Prop osition 3. If initial ly al l do ors ar e e qual ly likely to hide the c ar and if the host is e qual ly likely to op en either do or when he has a choic e, then, given the do or initial ly chosen and the do or op ene d, switching gives the player the 6 Car P1 Goat P2 Prize Figure 1: A Graphical Mo del (Bay es Net) for MHP c ar with c onditional pr ob ability 2 / 3 , whatever do or was initial ly chosen and which do or was op ene d (Morgan et al., 1991a,b). Pr o of. Prop. 1: This implication is trivial once w e observe that a “switcher” wins if and only if a “sta y er” loses. Prop. 2: W e use Ba y es’ theorem in the form p osterior o dds e quals prior o dds times likeliho o d r atio. The initial o dds that the car is b ehind do ors 1, 2 and 3 are 1:1:1. The p osterior o dds are therefore prop ortional to the probabilities that the host op ens Do or 3 giv en the play er chose Do or 1 and the car is b ehind Do or 1, 2 and 3 resp ectiv ely . These probabilities are q , 1 and 0 resp ectiv ely , where q = Prob( Host op ens Do or 3 | Play er chose Do or 1, car is b ehind same ) . The p osterior o dds for switching versus sta ying are therefore 1 : q , so that sta ying do es not hav e an adv antage ov er switc hing, whatev er q migh t b e. Since all do ors are initially equally likely to hide the car, the do or chosen b y the play er hides the car with probability 1/3. The unconditional prob- abilit y that switching gives the car is therefore 2/3. By the law of total 7 probabilit y , this can b e expressed as the sum ov er all six conditions (do or c hosen by play er, do or op ened by host), of the probability of that condi- tion times the conditional probability that switching giv es the car, under the condition. Eac h of these conditional probabilities is at least 1/2. The strat- egy of alwa ys switc hing can’t b e b eaten, since the success probability of an y other strategy is obtained from the success probabilit y of alwa ys switching b y replacing one or more of the conditional probabilities of getting the car b y switching by probabilities which are never larger. Prop. 3: If all do ors are equally lik ely to hide the car then b y indep endence of the initial choice of the pla y er and the lo cation of the car, the probability that the initial choice is correct is 1 / 3. Hence the unconditional probabilit y that switc hing giv es the car is 2 / 3. If the play er’s initial c hoice is uniform and the tw o probabilit y distributions in volv ed in the host’s c hoices are uni- form, the problem is symmetric with resp ect to the num b ering of the three do ors. Hence the conditional probabilities w e are after in Prop osition 3 are all the same, hence by the la w of total probability equal to the unconditional probabilit y that switching gives the car, 2/3. Prop osition 3 also follows from the insp ection of the p osterior o dds com- puted in the pro of of Prop osition 2. On taking q = 1 / 2, the p osterior o dds in fa vour of switching are 2:1 (Morgan et al., 1991a). In the literature, Prop osition 3 is usually prov en by explicit computation or tabulation, i.e., b y going bac k to first principles to compute the conditional probabilit y in question. F or instance, Morgan et al. (1991a) also give this direct computation, attributing it to Mosteller’s (1965) solution of the Pris- oner’s dilemma paradox. It is often offered as an example of Ba yes’ theorem, but really is just an illustration of conditional probabilit y via its definition. On the other hand, Ba y es’ theorem in its o dds form (whic h I used to pro ve Morgan et al.’s Prop osition 2) is a genuine the or em , and offers to m y mind a muc h more satisying route for those who like to see a computation and at the same time learn an imp ortan t concept and a p ow erful to ol. T o my mathematical mind the most elegan t pro of of Prop osition 3 is the argument b y symmetry starting from Proposition 1: the conditional probability is the same as the unconditional since all the conditional probabilities m ust b e the same. I learnt this pro of from Boris Tsirelson on wikip edia discussion pages, but it is also to b e found in Morgan et al. (1991b). This pro of also supplies one reason why the literature is so confused as to what constitutes a solution to MHP . One could apply symmetry at the outset, to argue that we only w an t an unconditional probability . There is no 8 p oin t in conditioning on an ything whic h w e can see in adv ance is irrelev ant to the question at hand. The pages of wikip edia, as well as a num b er of pap ers in the literature, are the scene of a furious contro v ersy mainly as to whether Prop osition 1 and a pro of thereof, or Prop osition 3 and a pro of thereof, is a “complete and correct solution to MHP”. These t w o solutions can be called the simple or p opular or unconditional solution, and the full or complete or conditional solution resp ectiv ely . The situation is further complicated b y the fact that many supp orters of the p opular solution do accept all the symmetry (uniformity) conditions of Prop osition 3, for a v ariet y of reasons. I will come bac k to this in the next main section, but first consider a rather differen t kind of result whic h can b e obtained within exactly the same general framework as b efore. 2.2 What game theory tells us Let us think of the four actions of the previous subsection as t wo pairs of mo ves in a tw o stage game b etw een the host and the play er in which the pla yer wan ts to get the car, the host w ants to k eep it. V on Neumann’s minimax theorem tells us that there exist a pair of minimax strategies for pla yer and host, and a v alue of the game, sa y p , having the follo wing defining c haracteristics. The minimax strategy of the pla y er (minimizes his maxim um c hance of losing) guarantees him at least probability p of winning, whatever the strategy of the host; the minimax stategy of the host (minimizes his maxim um probabilit y of losing) guaran tees him at most probability p of losing. If both pla y er and host pla y their minimax strategy then indeed the pla yer will win with probability p . Prop osition 4. The player’s str ate gy “initial choic e uniformly at r andom, ther e after switch” and the host’s str ate gy “hide the c ar uniformly at r andom, op en a do or uniformly at r andom when ther e is a choic e” form the minimax solution of the finite two-p erson zer o-sum game in which the player tries to maximize his pr ob ability of getting the c ar, the host tries to minimize it. The value of the game is 2 / 3 . Pr o of. W e must v erify t w o claims. The first is that whatev er strategy is used b y the host, the pla y er’s minimax strategy guaran tees the play er a success c hance of at least 2/3. The second is that whatev er strategy is used by the play er, the host’s minimax strategy prev ents the play er from achieving a success c hance greater than 2/3. F or the first claim notice that if the play er chooses a do or uniformly at random and thereafter switc hes, he’ll get the car with probability exactly 2/3; that follo ws from Prop osition 1. 9 F or the second, supp ose the host hides the car uniformly at random and thereafter op ens a do or uniformly at random when he has a choice. Making the initial c hoice of do or in an y wa y , and thereafter switc hing, giv es the pla y er success chance 2/3, and by Prop osition 2 (or 3, if you prefer) there is no wa y to impro ve this. Note that I did not use von Neumann’s theorem in any w ay to get this result: I simply made use of the probabilistic results of the previous subsec- tion. MHP w as brough t to the attention of the mathematical economics and game theory comm unity by a pap er of Nalebuff (1987), which contains a n umber of game-theoretic or economics c hoice puzzles. He considered MHP as an amusing problem with whic h to while aw ay a few min utes during a b oring seminar. After describing the problem he very briefly repro duced the short solution corresp onding to Prop osition 1. He enigmatically drops the names of Neumann-Morgenstern and Ba y es as he p onders why most p eople in real life to ok the wrong decision, but he do es not waste any more time on MHP . V ariants of the MHP in whic h the host do es not alwa ys open a door, or where he might b e trying to help you, or might b e trying to cheat you, lend themselv es very well to game theoretic study , see wikip edia or Rosenhouse (2009) for references. F or presen t purp oses, the imp ortant p oin t which I think is brought out b y a game theoretic approac h is that the play er do es ha ve tw o decision mo- men ts. The play er has control o ver his initial choice. V os Sav an t describ es the situation at the momen t the play er m ust make up his mind whether to switc h or stay , and most, but not all, p eople will instinctively feel that this is the only imp ortan t decision moment. But the pla yer earlier had a chance to c ho ose any do or he lik ed. Perhaps he would hav e been wise to think about what he would do if he did mak e it to the last round of the show, b efore setting off to the TV studio. There is no w ay he can ask the advice of a friendly mathematician as he stands on the stage under the dazzling studio ligh ts while the audience is shouting out conflicting advice. V an Damme (1995; in Dutc h) go es a little deeper in to the question of wh y real h uman play ers did not b ehav e rationally on the Mon ty Hall sho w; this is one of the main questions studied in the psychology , philosophy , artificial in telligence and animal b eha viour literature on MHP . Since “rational exp ec- tations” pla y a fundamental role in modern economic theory , the actual facts of the real world MHP , where play ers almost never switched do ors, b o des ill for the application of economic theory to real world economics. The usual rationale for human rational exp ectations in economics is that h umans learn 10 from mistak es. Ho wev er, the same p erson did not get to pla y several times times in the final round of the Mont y Hall sho w, and apparen tly no-one k ept a tally of what had happ ened to previous con testants, so learning simply did not take place. Nob o dy thought there would b e a p oint in learning! Instead, the pla yers used their brains, came to the conclusion that there was no ad- v antage in switching, and mostly stuc k to their original choice. At this p oin t they do make a rational choice: there w ould b e a muc h larger emotional loss to their ego on switc hing and losing, than on staying and losing. Sticking to y our do or demonstrates moral fortitude. Switc hing is feckless and deserves punishmen t. In terestingly , pigeons (sp ecifically , the Ro ck Pigeon, Columb a Livia , the pigeon whic h tourists feed in city squares all o ver the world) are v ery go o d at learning how to win the Mont y Hall game, see Herbranson and Schroeder (2010). They do not burden their little brains thinking ab out what to do but just go ahead and c ho ose. There is a lot of v ariation in their initial decisions whether to switc h or stay , and observing the results giv es them a c hance to learn from the past without thinking at all. Only a very small brain is needed to learn the optimal strategy . And these birds are ev olutionarily sp eaking v ery succesful indeed. 3 Whic h assumptions? Prop ositions 1, 2 and 3 tell us in differen t wa ys that switc hing is a goo d thing. Notice that the mathematical conditions made are successively stronger and the conclusion dra wn is successiv ely stronger to o. As the conditions get stronger, the scop e of application of the result gets narrow er: there are more assumptions to b e justified. F rom a mathematical p oin t of view none of these results are stronger than an y of the others: they are all strictly differ ent . The literature on MHP fo cusses on v ariants of Prop osition 1, and of Prop osition 3. These corresp ond to what are called the p opular or simple or unconditional solution, and the full or conditional solution to MHP . The full solution is mainly to b e found in in tro ductory probability or statistics texts, whereas the simple solution is p opular just ab out ev erywhere else. The in termediate “Prop osition 2” is only o ccasionally men tioned in the literature. The full list of conditions in Prop osition 3 is often called, at least in the kind of texts just mention ed, the standard or canonical or original MHP . I will just refer to them as the c onventional supplementary assumptions . Regarding the word “original”, it is a historical fact that Selvin (1975a) ga ve MHP its name, did this in a statistics journal, and wrote do wn the con ven tional full list of uniformity assumptions. He pro ceeded to compute 11 the unc onditional probabilit y that switc hing gives the car by en umeration of nine equally likely cases, for whic h he to ok b oth the play er’s initial choice and the lo cation of the car as uniform random, and of course indep enden t of one another. In his second note, Selvin (1975b), he computed the c ondi- tional probability using no w his full list of assumptions concerning the host’s b eha viour, and fixing the initial choice of the pla y er, but without noting any conceptual, let alone tec hnical, difference at all with his earlier solution. Of course, the n um b er “2/3” is the same. In the same note he quotes with appro v al from a letter from Mon ty Hall himself who gav e the argument of Prop osition 1: switching gives the car with probabilit y 2/3 b ecause the initial c hoice is righ t with probabilit y 1/3. W e know Mon ty will op en a do or re- v ealing a goat. Conditioning on an even t of probability one do es not change the probabilit y that the initial choice w as righ t. Th us Selvin set the seeds for subsequent confusion. Let me call his ap- proac h the pr actic al-minde d appr o ach : The right answer to MHP is “2/3”. There are many w a ys to get to this answer, and I am not to o muc h concerned how you get there. As long as y ou get the righ t answ er 2/3, w e’re happ y . After all, the whole point of MHP is that the initial instinct of ev ery one hearing the problem is to sa y “since the host could anyw ay op en a do or sho wing a goat, his action do esn’t tell me anything. There are still t wo do ors closed so they still are equally likely to hide the car. So the probability that switc hing would giv e the car is “1/2”, so I am not going to switch, thank y ou. Selvin’s t w o pap ers together gav e MHP a firm and more or less standard p osition in the elementary statistics literature. There is a con ven tional com- plete sp ecification of the problem. This enables us to write down a finite sample space and allo cate a probabilit y to every single outcome. Usually the pla yer’s initial c hoice is taken, in the light of the other assumptions without loss of generality , as fixed. All randomness is in the actions of the host, or in our lack of any know le dge ab out them. This corresponds to whether the writer has a frequen tist or a sub jectivist slan t, often not explicitly stated, but implicit in ve rbal hin ts. The question is not primarily “should you switch or sta y?”, but “what is the probability , or your probability , that switc hing will giv e the car?” Typically , as in Selvin’s second, conditional, approach, the pla yer’s initial choice is already fixed in the problem statement, the host’s t wo actions are already seen as completely random, whether b ecause we are told they are, ob jectively , or b ecause we are completely ignorant of how they are made, sub jectively . The problem t ypically features in the chapter which 12 in tro duces conditional probabilit y and Ba y es’ theorem in the discrete setting. Th us the problem is p osed b y a maths teac her who wan ts the student to learn conditional probabilit y . The problem is further reformulated as “what is the c onditional probabilit y that switc hing will give the car”. In suc h a con text not m uc h attention is b eing paid to the meaning of probabilit y . After all, right now w e are just busy getting accustomed to its calculus. Most of the examples figure pla ying cards, dice and balls in urns, and so the probability spaces are usually completely sp ecified (all outcomes can b e en umerated) and mostly they are symmetric, all elemen tary proba- bilities are equal. The studen t is either supp osed to “kno w”, or he is told explicitly , that the car is initially equally likely to b e b ehind any of the three do ors. The host is supposed to choose at random (shorthand for uniformly , or completely , at random) when he has a choice. Since these facts are given or supp osed to be guessed, the initial choice of the play er is irrelev ant, and w e are indeed alw a ys told that the pla y er has already pick ed Do or 1. W ell, if MHP is merely an exercise in conditional probability where the mathematical mo del is sp ecified in adv ance by the teacher, then it is clear ho w w e are to pro ceed. But I prefer to tak e a step back and to imagine you ar e on a game show . How could we “know” these probabilities? This is esp ecially imp ortant when one has the task of “explaining” MHP to non- mathematicians and to non-statisticians. This is where philosophy , or if y ou prefer, metaphysics, raises its head. Ho w can one “kno w” a probabilit y; what do es it mean to “know” a proba- bilit y? I am not going to answer those questions. But I am going to compare a con ven tional frequen tist view – probabilit y is out there in the real w orld – to a con ven tional Bay esian view – probability is in the information which w e p os- sess. I hop e to do this neutrally , without taking a dogmatic stance myself. It is a fact that man y amateur users of probability are instinctive sub jectivists, not so many are instinctiv e frequentists. Let us try to work out where either instinct w ould tak e us. An imp ortant thing to realise is: Ba yesian probabil- ities in, Bay esian probabilities out; frequentist in, frequentist out. I will also emphasize the difference which comes from seeing randomness in the host’s moves or in the player’s moves , and the difference whic h comes from seeing the question as asking for an action , or more passively for a pr ob ability . F or a sub jectivist (Ba yesian) the MHP is very simple indeed. W e only kno w what w e hav e b een told b y vos Sav ant (Whitaker). The wording “ say , Do or 1” and “ say , Do or 3” (my italics) emphasize that we know nothing ab out the b ehaviour of the host, whether in hiding cars or in op ening do ors. Kno wing nothing, the situation for us is indistinguishable from the situation in which w e had b een told in adv ance that car hiding and do or op ening was 13 actually done using p erfect fair randomizers (unbiased dice throws or coin tosses). Probability is a represen tation of our uncertain knowledge about the single instance under consideration. Probabilit y calculus is the unique in ter- nally consisten t wa y to manipulate uncertain knowledge. T o start off with, since w e know nothing, we ma y as w ell choose our do or initially according to our p ersonal luc ky num b er, so we pick ed Do or 1. Having seen the host op en Do or 3, we w ould now b e prepared to b et at o dds 2:1 that the car is b ehind Do or 2. The new situation is indistinguishable for us from a b etting situation with fair o dds 2:1 based on a p erfect fair randomizer (by whic h I simply refer to the kind of situation in which sub jectivists and ob jectivists tend to agree on the probabilities, even if they think they mean something quite differen t). Do es the Bay esian (a sub jectivist) need Ba yes’ theorem in order to come to his conclusion? I think the answer is no . F or a sub jectivist the do or num- b ers are irrelev an t. The problem is unchanged by ren um b ering of the do ors. His b eliefs ab out whether the car w ould be b ehind the other do or in any of the six situations (do or c hosen, do or op ened) would b e the same. He has no need to activ ely compute the conditional probabilit y in order to confirm what he already kno ws. He could use Prop osition 3 but is only interested in Prop osition 1. The symmetry argument of my pro of of Prop osition 3 is the mathematical expression of his prior knowledge that he may ignore the do or num b ers and just compute an unconditional probability . Do you notice the symmetry in adv ance and take adv an tage of it, or just compute aw a y and notice it afterwards? It do esn’t matter. The answer is 2 / 3 and it is a conditional and unconditional probabilit y at the same time. What is imp ortan t to realise is that the probability computed b y a sub- jectivist is also a sub jective probability . Starting from probabilities whic h ex- press prior personal expectations, the probability we ha ve deriv ed expresses ho w our prior p ersonal exp ectations as to the lo cation of the car should logically b e mo dified on seeing the host op en Do or 3 and reveal a goat in resp onse to our c hoice of Do or 1. These probabilities sa y nothing ab out what w e w ould exp ect to see if the game was rep eated many times. W e might well exp ect to see a systematic bias in the lo cation of the car or the choice of the host. Our uniform prior distributions express the fact that our prior b eliefs or prior information ab out such biases are in v ariant under p ermutations of the do or n um b ers. F or a frequentist, MHP is harder – unless the problem has already b een mathematized, and the frequen tist has b een told that the car is hidden com- pletely at random and the host c ho oses completely at random (when he has a c hoice) to o. P ersonally , I don’t find this a v ery realistic scenario. I can think of one semi-realistic scenario, and that is the scenario prop osed b y 14 Morgan et al. (1991a). Supp ose we hav e inside information that every week, the car is hidden uniformly at random, in order to mak e its lo cation totally unpredictable to all play ers. How ev er Mon ty’s choice of do or to op en, when he has a c hoice, is something which go es on in his head at the spur of the momen t. In this situation w e may as well let our initial choice of do or b e determined b y our luc ky n um b er, e.g., Do or 1. Prop osition 2 tells us that not only is alw ays switching a wise strategy , it tells us that we cannot do b etter. No need to worry our heads ab out what is the conditional probabilit y . It is nev er against switching. There is just one solution whic h do es not require any prior knowledge at all; instead it requires prior action. T aking our cue from the game theoretic solution, we realize that the pla yer has t wo opp ortunites to act, not one. W e allo w ourselves the latitude to reform ulate v os Sav an t’s w ords as “Y ou are going to b e on a game show...”. W e advise vos Sav an t, or her corresp ondent Craig Whitak er, to tak e fate in to his or her o wn hands. Before the sho w, pic k your lucky num b er (1, 2 or 3) b y a toss of a fair die. When you make it to the final round, choose that do or and thereafter switch. By Prop osition 1 y ou’ll come home with the car with probabilit y 2/3, and by Prop osition 4 that’s the b est y ou can hop e for. Both frequen tists and sub jectivists will agree that you win the car with probabilit y 2/3 in this w ay . They will likely disagree ab out whether the conditional probability that you win, given do or chosen and do or op ened, is also 2/3. I think the frequentist will say that he do es not kno w it since he do esn’t kno w anything about the tw o host actions, while the sub jectivist will sa y that he do es know the conditional probabilit y (and it’s 2/3) for the v ery same reason. So what? 4 Conclusion The Mont y Hall Problem offers muc h more to the student than a mindless exercise in conditional probability . It also offers a challenging exercise in mathematical mo delling. I notice three imp ortan t lessons. (1) The more you assume, the more you can conclude, but the more limited are your conclu- sions. The honest answer is not one mathematical solution but a range of solutions. (2) Whether y ou are a sub jectivist or a frequentist affects the ease with whic h you migh t mak e probabilistic assumptions but simultaneously af- fects the meaning of the conclusions. (3). Think out of the b o x. V os Sav an t asks for an action , not for a pr ob ability . The pla y er has two decision momen ts during the sho w, not one. 15 References Burns, B. D. and Wieth, M. (2004), The collider principle in causal reasoning: wh y the Mon t y Hall dilemma is so hard, J. Exp erimental Psycholo gy: Gener al 133 434–449. http://www.psych.usyd.edu.au/staff/bburns/Burns Wieth04 man.pdf v an Damme, E. E. C. (1995), Rationaliteit. Ec on. Stat. Berichten 15–11–1995, 1019. Herbranson, W. T. and Sc hro eder, J. (2010), Are birds smarter than mathematicians? Pigeons (Colum ba livia) perform optimally on a v ersion of the Mon t y Hall Dilemma. J. Comp. Psychol. 124 1–13. http://people.whitman.edu/ ∼ herbrawt/HS JCP 2010.pdf Hogbin, M. and Nijdam, W. (2010), Letter to the editor. Am. Statist. 64 193. Morgan, J. P ., Chagant y , N. R., Dahiy a, R. C., and Doviak, M. J. (1991a), Let’s mak e a deal: The play er’s dilemma. A m. Statist. 45 284–287. Morgan, J. P ., Chagant y , N. R., Dahiy a, R. C., and Doviak, M. J. (1991b), Rejoinder to Seymann’s commen t on “Let’s make a deal: the play er’s dilemma”. A m. Statist. 45 289. Morgan, J. P ., Chagant y , N. R., Dahiy a, R. C., and Doviak, M. J. (2010), Resp onse to Hogbin and Nijdam’s letter, Am. Statist. 64 193–194. Mosteller, F. (1965), Fifty Chal lenging Pr oblems in Pr ob ability with Solutions . Do ver, New Y ork. Nalebuff, B. (1987), Puzzles: Cho ose a curtain, duel-ity , tw o p oint con versions, and more, J. Ec on. Persp e ctives 1 (2) 157–163. Rosenhouse, J. (2009), The Monty Hal l Pr oblem , Oxford Univ ersit y Press. Rosen thal, J. S. (2005), Struck by Lightning: The Curious World of Pr ob abilities . Harp er Collins, Canada. Rosen thal, J. S. (2008), Mont y Hall, Mon ty F all, Mon t y Crawl, Math Horizons 16 5–7. http://probability.ca/jeff/writing/montyfall.pdf Selvin, S. (1975a), A problem in probability (letter to the editor). Am. Statist. 29 67. Selvin, S. (1975b), On the Mon t y Hall problem (letter to the editor). Am. Statist. 29 134. 16 Seyman, R. G. (1991), Commen t on “Let’s mak e a deal: the play er’s dilemma”. A m. Statist. 64 287–288. 17