The Three Doors Problem...-s

The Three Do ors Problem... - s ∗ Ric hard D. Gil l Mathematica l Institute, Univ ersit y Leiden, Netherlands http://w ww.math. leidenuniv.nl/ ∼ gill Marc h 1, 20 1 0 1 In tro du ction The Thr e e Do ors Pr oblem , or Monty Hal l Pr oblem , is familiar to statisticians as a parado x in elemen tary probabilit y theory often found in ele men tary probabilit y texts (esp ecially in their exercises sections). In that contex t it is usually mean t to b e solv ed b y careful (and elemen ta ry) a pplication of Ba yes ’ theorem. How ev er, in differen t forms, it is m uc h discussed and argued ab out and written ab o ut b y psyc holo g ists, game-t heorists and mathematical economists, educ ationalists, journalists, la y p ersons , blog- writers, wikip edia editors. In this a rticle I will briefly surv ey the history of t he problem and some of the appro ac hes to it whic h hav e b een prop osed. My take-home message t o y ou, dear reader, is that one should distinguish t wo lev els to the problem. There is an info rmally stated pro blem whic h y ou could p ose to a friend at a part y; and there are man y concrete versions or r e alizations of the problem, whic h are actually the result of mathematical or probabilistic or statistical mo del ling . This mo delling often in volv es adding supplemen tary assumptions c hosen to ma ke the problem well p osed in the terms of the mo deller. The mo deller finds those assumptions p erfectly natura l. His or her studen ts ar e supp osed to guess those assumptions from v arious k ey words (lik e: “indis- tinguishable”, “unkno wn”) strategically placed in the problem re-statemen t. T eac hing statistics is of ten ab out teaching the studen ts to read the teacher’s mind. Mathematical (proba bilistic, statistical) mo delling is, unfortunately , often solution driv en r a ther t ha n problem drive n. ∗ v.2 // arXiv.or g:1002 .3878 [stat.AP] // Submitted to Springer L ex ic on of Statistics 1 The v ery same criticism can, and should, b e lev elled at this v ery article! By cunningly presen ting the history of The T hr e e Do ors Pr oblem from m y rather special p oin t of v iew, I ha ve engineered complex reality so as to conv ert the Th r e e Do ors Pr oblem into an illustration of m y p ersonal Philosophy o f Science, m y Philosoph y o f Statistics. This means that I ha v e r e- engineered the Thr e e D o ors Pr oblem in to an example of the p o int of view that Applied Statisticians should alw a ys b e w ary of the lure of S o lution-driven Scienc e . Applied Statisticians are tr a ined to kno w Applied Statistics, and are trained to kno w how to con vert real w orld problems in to statistics problems. That is fine. But the b est Applied Statis- ticians kno w that Applied Statistics is not the only game in town. Applied Statisticians are merely some particular kind of Scien tists. They kno w lo ts ab out mo delling uncertaint y , and ab out learning from more or less ra ndo m data, but probably not muc h ab out an ything else. The Real Scien tist kno ws that there is not a univ ersal disciplinary approach to ev ery pro blem. The R e al Statistic al Scien tist mo destly and p ersuasiv ely and realistically offers what his or her discipline has t o offer in synergy with o thers. T o summarize, w e m ust distinguish b et w een: (0) the Thr e e-Do ors-Pr oblem Pr oblem [sic], which is to mak e sense of some real world question of a real p erson. (1) a large n um b er of solutions to this meta -problem, i.e., the many Three- Do ors- Problem Pr oblems , whic h are comp eting mathematizations of the meta- problem (0 ) . Eac h of the solutions at lev el (1) can well ha ve a n um b er of different solutions: nice o nes and ugly ones; correct ones and incorrect ones. In this article, I will discuss three leve l (1) solutions, i.e., three differen t Mon t y Ha ll problems; and try t o give three short correct and attractiv e solutions. No w read on. Be critical, use y our in tellect, don’t b eliev e any thing on authorit y , and certainly not on mine. Esp ecially , don’t forget the problem at meta-meta-lev el ( − 1), not listed ab ov e. C’est la vie . 2 Starting P oin t I shall start not with the historical r o ots of the problem, but with the question whic h made the Three Do ors Problem fa mous, ev en reaching the fro nt page of the New Y ork Times . Marilyn v os Sa v an t (a w o ma n allegedly with the highest IQ in the w orld) p osed the Thr e e Do or Pr oble m or Monty Hal l Pr oblem in her “ Ask Marilyn” 2 column in Par ade magazine (Septem b er, 1990, p. 16 ), as p osed to her by a corresp onden t, a Mr. Craig Whitak er. It w as, quoting v os Sav an t literally , the fo llo wing: Supp ose you’r e on a game show , and you’r e given the choic e of thr e e do ors: Behind on e do or is a c ar; b ehind the o thers, go ats. Y ou pic k a do or, sa y No. 1, and the host, who kno w s what’s b ehind the d o ors, op ens ano ther do or, say No. 3, whic h 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 ch oic e? Apparen tly , the problem refers to a real American TV quiz-sho w, with a real presen ter, called Mon t y Hall. The literature on the Mont y Hall Problem is enormous. At the end of t his article I shall simply list tw o references whic h for me hav e b een esp ecially v aluable: a pap er b y Jeff Rosenthal (200 8 ) and a b o ok by Jason Rosenhouse (2009). The latt er has a h uge r eference list and discusses the pre- and p ost- history of v os Sav ant’s problem. Briefly regarding the pre-history , one may trace the pro blem bac k through a 197 5 letter t o the editor in the journal The A meric an Statistician b y bio- statistician Steve Selkin, to a problem called The Thr e e Prisoners Pr oblem p osed by Stephen Gardner in his Mathematical Games column in Scientific A meric an in 195 9 , and from there back to Bertr and’s B ox Pr oblem in his 1889 text on Probabilit y Theory . The in ternet encyclop edia wikipedia.org discussion pages (in man y languag es) are a fabulous though every -c ha ng ing resource. Almost ev erything that I write here w as learn t f r o m t ho se pages. Despite making homage here to the tw o cited a uthors Rosen thal (20 0 8) and Rosenhouse (2009) for their w o nderful work, I emphasize that I strongly disagree with b o th R o senhouse (“the canonical problem”) and Ro sen thal (“the original problem”) on what the essen tial Mon t y Hall problem is. I am more angry with certain other authors, who will remain nameless but for the sak e of argumen t I’ll just call Morgan et al. , for unilaterally declaring in The Americ an S tatistician in 1 9 81 their Mon t y Hall problem to b e t he only p ossible sensible problem, for calling ev ery one who solv ed different problems stupid, and for getting an incorr ect theorem 1 published in the p eer-review ed literature. Deciding unilat erally (Rosenhouse, 2009) that a certain form ulation is c anonic al is asking for a sc hism and for excomm unication. Calling a particu- lar v ersion original (Rosen thal, 2008) is asking fo r a historical con tra diction. 1 I refer to their result ab out the situation when we do not know the quiz-master ’s probability of op ening a pa rticular do or when he has a choice, and put a uniform pr ior on this probability . 3 In view of the pre-history of t he problem, the notion is not well defined. Mon ty Hall is par t o f folk-culture, culture is aliv e, the Mon ty Hall problem is not owned b y a pa r t icular kind o f mathematician who lo oks at such a problem fro m a particular p oint of view, and who adds f or them “ natural” extra assumptions whic h merely ha ve t he role o f allo wing their solution to w ork. Presen ting an y “canonical” or “original” Mon ty Hall problem t o gether with a solution, is an example of solution driven scienc e — you hav e learn t a clev er trick and w ant to show that it solv es lots of problems. 3 Three Mon t y Hall Problems I will concen trate on three differen t particular Mont y Hall problems. One of them (Q-0) is simply to answ er the question literally p osed b y Marilyn v os Sa v ant, “w ould you switc h?”. The other t wo (Q-1, Q-2) are p opular math- ematizations, particularly p opular among exp erts or teac hers of elemen tary probabilit y theory: o ne asks for the unconditiona l probability that “alw a ys switc hing” would gets the car, the o t her asks for the conditional probability giv en t he choice s made so far. Here they are: Q-0: Marilyn v o s Sav a nt’s (or Craig Whitaker’s ) question “ Is it to your ad- vantage to switch? ” Q-1: A mathematician’s question“ What is the unc onditional p r ob abil i ty that switching g i v es the c ar? ” Q-2: A mathematician’s question “ What is the c onditional pr ob ability that switching g i v es the c ar, given everything so fa r? ” The free, and fr eely editable, in ternet encyclopedia Wikip edia is the scene o f a furious debate a s to whic h mathematization Q-1 or Q-2 is the righ t start- ing p oint for answ ering the v erbal question Q-0 (to b e honest, many of the actors claim another “or iginal” question as the original question). Along- side t ha t, t here is a furious debate as to whic h supplemen ta ry conditions are ob viously implicitly b eing made. F or eac h protag o nist in the debate, those are the a ssumptions whic h ensure that his or her question has a unique a nd nice answ er. My own h umble o pinion is “neither Q-1 nor Q-2, though the unconditional approac h comes closer’. I prefer Q-0, and I prefer to see it as a question of game the ory for whic h, to my mind, [almost] no supplemen tary conditions need to b e made. Here I admit that I will supp ose that the pla y er kno ws game-theory and came to the quiz-sho w prepared. I will also supp ose that the play er w ants to get the Cadillac while Mont y Hall, the quizmaster, w ants to k eep it. 4 My analysis b elow o f b oth problems Q-1 and Q-2 yields the go o d an- sw er “2 / 3” under minimal assumptions, and almost without computation or algebraic manipulation. I will use Israeli (formerly So viet Union) mathemati- cian Boris Tsirelson’s prop o sal on Wikip edia ta lk pag es to use symmetry t o deduce the conditional probability from the unconditional one. (Boris gra- ciously ga v e me p ermission to cite him here, but this should not b e in terpreted to mean that anything written here also has his a pprov al). Y ou, the r eader, ma y w ell prefer a calculation using Bay es’ theorem, or a calculation using the definition of conditional probabilit y; I think this is a matter of ta ste. I finally use a game-theoretic p o int of view, and von Neumann’s minimax theorem, to answ er the question Q -0 p osed by Marilyn v os Sav ant, on the assumptions just stated. Let the three doo r s b e num b ered in adv ance 1, 2, and 3. I add the univ ersally ag r eed (and historically correct)additio na l assumptions: Mon ty Hall kno ws in adv ance where the car is hidden, Mont y Hall alw ays op ens a do or rev ealing a goat. In tro duce four random v ariables ta king v alues in the set of do or- n umbers { 1 , 2 , 3 } : C : the quiz-team hides the Car (a Cadillac) b ehind do or C , P : the Pla y er choo ses do or P , Q : the Quizmaster (Mon t y Hall) op ens do o r Q , S : Mon ty Hall asks the play er if she’ld like to Switc h to do or S . Because of the standard story of the Mon ty Hall sho w, w e certainly ha v e: Q 6 = P , the quizmaster always op ens a do o r differen t to the play er’s first choice, Q 6 = C , op ening that do or always reve als a goat, S 6 = P , the play er is always inv ited to switc h to another do or, S 6 = Q , n o play er w an ts to go home with a goat. It do es not matter fo r the subsequen t mathematical a nalysis whether pro ba- bilities are sub jectiv e ( Bay esian) or ob jectiv e ( f requen tist); nor do es it matter whose probabilities they are supp o sed to b e, at what stage of the game. Some writers think of the play er’s initia l c ho ice a s fixed. F or them, P is degenerate. I simply merely down some mathematical assumptions a nd deduce math- ematical consequences of them. 5 4 Solutio n to Q-1: uncondi tional c hanc e t hat s witc h ing wins By the rules of t he ga me and the definition of S , if P 6 = C then S = C , and vice-v ersa. A “switc her” w ould win the car if and only if a “ stay er” w ould lose it. Therefore: If Pr( P = C ) = 1 / 3 then Pr ( S = C ) = 2 / 3 , sinc e the two events ar e c om p lementary. 5 Solutio n to Q-2: probability car is b ehind do or 2 giv en y ou c hose do or 1, Mon t y Hall op ened 3 First of all, supp ose that P and C are uniform and indep endent, and that giv en ( P , C ), supp ose that Q is unifor m on its p ossible v alues (unequal to those of P and of C ). Let S b e defined as b efore, as the third do or- n umber differen t from P and Q . The join t law of C , P , Q, S is b y this definition in v ariant under renum b erings of the three do ors. Hence Pr ( S = C | P = x, Q = y ) is the same for all x 6 = y . By the la w of to t a l probability , Pr( S = C ) (whic h is equal to 2 / 3 b y o ur solution to Q-1) is equal to the w eighted av erage of all Pr( S = C | P = x, Q = y ), x 6 = y ∈ { 1 , 2 , , 3 } . Since the latter are all equal, all these six conditiona l pro babilities are equal to their a v erage 2 / 3 . Conditioning on P = x , sa y , and letting y and y ′ denote the r emaining t wo do or n um b ers, w e find the follow ing corollary: No w tak e the do or c hosen b y the pla yer as fixed, P ≡ 1, sa y . W e are to compute Pr( S = C | Q = 3). Assume that al l do ors ar e e qual ly likely to hide the c ar a n d assume that the quizmaster cho oses c ompl e tely at r andom when he has a choic e . Without loss of generalit y we ma y as w ell pretend that P was chose n in adv ance completely at random. No w w e hav e embedded our problem into the situation just solv ed, where P and C are uniform a nd indep enden t. If P ≡ 1 is fixe d, C is uniform, and Q is symmetric, then “sw itch- ing gives c ar” is ind e p endent of quizmaster’s choic e, henc e Pr( S = C | Q = 3) = Pr( S = C | Q = 2 ′ ) = Pr ( S = C ) = 2 / 3 . Some readers may prefer a direct calculation. Using Ba ye s’ theorem in the form “ p osterior o dds equal prior o dds times lik eliho o ds” is a particularly 6 efficien t w ay to do this. The pro babilities a nd conditional probabilities b elow are all conditional on P = 1, or if y our prefer with P ≡ 1. W e hav e uniform prior o dds Pr( C = 1) : Pr( C = 2) : Pr( C = 3) = 1 : 1 : 1 . The lik eliho o d for C , the lo cation of the car, given da ta Q = 3, is (prop or- tional to) the discrete densit y function of Q giv en C (a nd P ) Pr( Q = 3 | C = 1) : Pr( Q = 3 | C = 2) : Pr( Q = 3 | C = 3) = 1 2 : 1 : 0 . The p osterior o dds are therefore prop ort io nal to the like liho o d. It follow s that the p osterior pr o babilities are Pr( Q = 3 | C = 1) = 1 3 , Pr( Q = 3 | C = 2) = 2 3 , Pr( Q = 3 | C = 3) = 0 . 6 Answ er to Marylin v os Sa v an t’s Q-0: should you switc h do o rs? Y es. Recall, Y ou only know that Monty Hal l always op en s a do or r eve aling a go at . Y ou didn’t know what strategy the quiz-team a nd quizmaster w ere going to use for their c hoices of the distribution of C and the distribution of Q g iven P and C , so naturally (since you kno w elemen tary Ga me Theory) y ou had pick ed y our do or uniformly at ra ndom. Y our strategy o f c ho osing C uniformly at random guarantee s that Pr ( C = P ) = 1 / 3 and hence that Pr( S = C ) = 2 / 3 . It w a s easy for y ou to find out that this com bined strategy , whic h I’ll call “symmetrize and switc h”, is your so-called minimax strategy . On the one ha nd, “symmetrize and switc h” guarantee s y o u a 2 / 3 (uncon- ditional) chance of winning the car, whatev er strat egy used by the quizmaster and his team. On the other hand, if the quizmaster and his team use their “symmetric” strategy “hide the car uniformly at ra ndom and toss a fair coin to op en a do or if there is choice ”, then y ou cannot win the car with a b etter probabilit y than 2 / 3. The fa ct that your “symmetrize and switc h” strategy giv es you “ at least” 2 / 3, while the quizmaster’s “symmetry” strategy preve n ts y ou from doing b etter, prov es that these ar e the resp ectiv e minimax strategies, and 2 / 3 is the game-theoretic v alue of this tw o-party zero-sum game. (Minimax strate- gies and the accompany ing “v alue” o f the g ame exist b y virtue of John v on Neumann’s (1929 ) minimax theorem for finite t w o-party zero-sum ga mes). 7 There is not m uch p oin t for you in w orrying ab out y our conditio na l prob- abilit y of winning conditional on y our specific initial c hoice and the sp ecific do or op ened by the quizmaster, sa y do ors 1 and 3 resp ectiv ely . Y ou don’t kno w t his conditional probability any w ay , since y ou don’t know the strategy used b y quiz-team and the quizmaster. (Ev en though y ou kno w probabilit y theory a nd game theory , they ma yb e don’t). How ev er, it is may b e comforting to learn, b y easy calculation, that if the car is hidden unifo r mly at random, then y our conditiona l probability cannot b e sm al ler than 1 / 2. So in that case at least, it certainly nev er hurts to switc h do or. 7 Discuss ion Ab o v e I tried to giv e short clear mathematical solutions to three mathemat- ical problems. Tw o of them w ere problems of elemen tary proba bility theory , the third is a problem of elemen tary game theory . As suc h, it in volv es not m uch more than elemen tary probability theory a nd the b eautiful minimax theorem of John v on Neumann (1928) . That a finite tw o-party zero- sum game has a saddle-p oint, or in other w ords, that the tw o parties in suc h a game hav e match ing minimax strat egies (if randomization is allow ed), is not ob vious. It seems to me that probabilists ought to kno w more ab out game theory , since ev ery o rdinary non-mathematician who hears a b out the prob- lem starts to w onder whether the quiz-master is trying to c heat the pla ye r, leading to an infinite regress: if I know that he kno ws that I kno w that.... I am told that the literature of mathematical economics and of game theory is full of Mon ty Hall examples, but no- one can giv e me a nice reference to a nice g ame-theoretic solution of the pro blem. Probably g ame-theorists lik e to ke ep t heir clev er ideas to themselv es, so as to mak e money from pla ying the game. Only losers write b o oks explaining ho w the reader could mak e money from game theory . It w ould certainly b e in teresting to in ves tigate more complex game-theoretic v ersions of the problem. If we tak e Mon t y Hall as a separate play er to the TV statio n, a nd not e that TV r a tings are probably help ed if nice pla y ers win while annoy ing pla yers lose, we lea ve elemen tary game theory and mu st learn t he theory of Nash equilibria. Then there is a so ciological or historical question: who “ o wns” the Mont y Hall problem? I think the answ er is ob vious: no-one. A b eautiful mathemat- ical paradox, once launc hed into the real w orld, liv es it own life, it ev o lv es, it is re-ev aluated b y generation after generation. This p oin t of view actually mak es me b eliev e that Question 0: would you switch is the r ig h t question, and no further informa t io n should b e giv en b ey ond the fact that you kno w 8 that the quizmaster kno ws where the car is hidden, and alw ays op ens a do or exhibiting a goat. Question 0 is a question y ou can ask a non-mathematician at a party , and if they ha ve no t heard of the problem b efore, t hey’ll giv e the wrong answ er (or rather, o ne of t he t w o wrong answ ers: no b ecause nothing is c hanged, or it d o esn ’t matter because it’s no w 50–50). My mother, who w as one o f T uring’s computers a t Bletc hley P ark during the w ar , but who had had almost no sc ho oling and in particular nev er learn t any mathematics, is the only p erson I know who immediately said: switch , by immediate in tu- itiv e consideration of the 1 00-do o r v arian t of the problem. The pro blem is a p ar adox since y o u can next immediately convinc e an yone (except lawy ers, as w as sho wn b y an exp erimen t in Nijmegen), that their initia l answe r is wrong. The mathematizations Questions 1 and 2 are not (in m y h um ble opinion!) the Mont y Hall problem; they are questions whic h probabilists might a sk, anxious to sho w off Bay es’ theorem or whatev er. Some p eople in tuitiv ely t r y to a nsw er Question 0 via Questions 1 and 2; that is natura l, I do admit. And sometimes p eople b ecome v ery confused when they realize that the answ er to Question 2 can only b e giv en its prett y answ er “2/3” under further conditions. It is interes ting ho w in the p edagogical mathematical literature, the further conditions are as it w ere held under your nose, e.g. b y sa ying “three identic al do ors”, or replacing Marilyn’s ”say , do or 1” by the more emphatic “do or 1 ”. It seems to me that adding into the question explicitly the remarks that the three do ors are equally lik ely to hide the car, and that when the quizmas- ter has a choice he secretly tosses a fair coin to decide, conv ert this b eautiful parado x in to a probability puzzle with little app eal any more to non experts. It also con ve rts the problem in t o one v ersion of the three prisoner’s para- do x. The three prisoners problem is isomorphic to the conditional probabilis- tic three do ors problem. I alw a ys f o und it a bit silly and not v ery in teresting, but p ossibly that problem t o o should b e approac hed f r o m a sophisticated game theoretic p oint of view. By the w ay , Marilyn v os Sa v an t’s original question is semantically am- biguous, tho ug h this might not b e noticed by a non-native English sp eak er. Are the men tioned do or n umbers, h uge painted nu m b ers on the front of the do ors a priori , or are w e just for con ve nience na m ing the do o r s by t he c hoices of the actors in our game a p osteriori . Marilyn stated in a lat er column in Par ade that she had originally b een thinking of the latter. Ho wev er, her own offered solutions are not consisten t with a single unam biguous f orm ulat io n. Probably she did not find the difference v ery in t eresting. This little article contains nothing new, and o nly almost trivial mathe- matics. It is a plea for futur e generations to preserv e the life of The T rue Monty Hal l p ar adox , and not let themselv es b e misled by proba bilit y purists who say “you must compute a conditional probability”. 9 References Gill, Ric ha r d D . (201 0), The one and only true Mon ty Hall prob em, submitted to Statistic a Ne erland i c a . arXiv.org:1 002.0651 [math.HO] Rosenhouse, Jason (2 009), The Monty Hal l Pr oblem , Oxford Univ ersit y Press. Rosen thal, Jeffrey S. (2008), Mon t y Hall, Mont y F all, Mon ty Craw l, Math Horizons , Septem b er 2008, 5 –7. Reprint: http://prob ability.ca/ jeff/writing/montyfa ll.pdf 10