"Why all the hairsplitting? Why do you Math geeks worry about these little tiny details when a little common sense will do more for one's analysis than all of the fancy proofs in the world, except maybe in a few cooked up cases you guys come up with just to make yourselves look clever?' Answer: Because common sense doesn't do all that some would like to think it does, and it often gets lost in the mazes created by the quirks of the language, and of our own emotional responses. Let's consider the following two circumstances.



  1. Some years back, there was a game show called "let's make a deal". On it, at one stage in a game, the contestant would be presented with three doors. Behind one of the doors was an expensive prize, and behind the other two were gag prizes (a mule pulling a cart would be one of them) telling the contestant that she had won nothing. First, the contestant would choose a door. Then Monty would tell her how lucky she was that she hadn't picked "this other door", showing her a gag prize (signifying loss) behind one of the other doors. "Would you like to stay with the door you have, or choose the other door?", he would ask.

    Could one not argue that she has no reason to prefer one over the other? Why should the prize be any likelier to be behind one door or another?


  2. Three royalist soldiers have been captured during the French revolution. All are scheduled to be executed in the morning, but in a rare fit of compassion, their captors have decided to spare the life on one of them. As a matter of cruel whim, however, they decide to tell the prisoners this, the prisoners being kept in seperate cells, without telling them which of the three will be spared. Each, based on what he knows, can only give himself a one in three chance of living to see the next sunset; he is, to the best of his knowledge, playing Russian Roulette with four bullets, and is left with an uneasy night's sleep pondering that thought.

    One of the prisoners (let's call him Alexandre) calls a guard over. "Psst, Antoine", he says, "I know that you can not tell me if my life will be spared, but surely there is no harm in telling me the name of one of the men who is to be killed, other than myself? I already know there has to be at least one, yes, so surely you aren't revealing any secrets by telling me this?" The guard agrees, and reveals that Claude won't need to worry about planning for his retirement. The prisoner thinks that he has put one over on Antoine. "Before, I knew nothing, and could only assign myself a one in three chance of survival. Now I know that Claude will soon be fertilizing the garden, and surely there is no reason why Francois would be any likelier to be executed than me, or I than him? So my probability of survival, based on what I know, is now 1/2, and the guard HAS told me something."

    But how can this be? Isn't this absurd? There HAD to be at least one name that the guard could give, and the same argument would have followed, mutus mutandis, had he said "Francois" instead of "Claude", if we accept the argument above, so how can the giving of that name convery any information at all?


The answer in the case of the second problem is "it can't". Alexandre's chances of survival, based on what he learned, are still just one in three. In seeing what went wrong with his argument, we will come to see that the two problems are, in fact, in some sense the same problem in disguise and that what many would call "common sense" is, in fact, in conflict with itself.

Think of Antoine, the guard, as being Monty Hall, and each of the prisoners as a door. Yes, sounds a little weird, but such are the correspondences. Look beyond each of the prisoners to the life that awaits him, and you will see guillotins behind two of them, and a happy home behind the third. Try to guess which one of our three living "doors" has the happy prize waiting for him as his fate? The guard has done nothing more in answering that question than Monty Hall did, and nothing less. The fact that we may expect different answers regarding the probability of a favorable income is nothing more than a superstition, a reflection of our differing psychological responses to living human beings and inanimate objects, as your probable amusement when I told you to think of the prisoners as "doors" shows (that, and of the linguistic confusion that arises because the word "winning" changes meaning a little bit). But push the common sense a little further, and I think you will see that the nature of a "door" tells one little about the nature of what lies beyond it. Our emotions mislead us, so let's do the calculations, and put our attitudes aside.


A very simple argument for the first problem, courtesy of Marilyn Vos Savant of Mensa, goes roughly like this:



  1. The contestant knows nothing about what is behind any doors left unseen, and so really only has two strategies to choose from: choose and stay put, and choose and then change one's choice after Monty asks


  2. One is choosing blindly among three doors, only one of which leads to a winning outcome, so one's initial choice has only a one in three choice of being a good one.


  3. If one chooses "choose and stay put", one wins if and only if one's first choice was correct - one loses 2/3 of the time.


  4. If one chooses "choose and then switch", then one wins if one originally chose a losing door (true 2/3 of the time) because Monty just showed one the other losing door and one can't end up choosing that one. If one chose the winning door initially (1/3 probability), one (of course) ends up switching over to the losing door. Thus, by choosing the proper strategy (change, then switch) one can double one's chances of success.


Let's apply Vos Savant's argument to the prisoner's case. One is trying to guess behind which of the three "living doors" (ie. the prisoners) the "prize" (ie. the fate of a long life) is hidden behind. Applying the "Monty Halls" criterion of Victory, let's say Alexandre tries to guess which of the three prisoners will get to see the leaves fall; "victory" in this sense means correctly guessing the name of the survivor, and so Alexandre follows the choose then switch strategy, as argued above. Full of hope, he starts by guessing his own name, but we know that is not a name that the guard will give, so after he switches, with certainty he will end up naming one of the other orisoners as the survivor. By the argument above, in the Monty Hall sense, he has a 2/3 chance of "winning", but as victory will involve his head dropping into a basket, it seems a very Pyrrhic victory indeed - and there is the linguistic mirage that leads us astray. While the words "victory" and "success' are usable in each case, in context they mean very different things.

There, Ms. Vos Savant would perhaps let us stop, but in a very self-defeating way when dealing with our skeptical extreme intuitionist, who will retort that this just goes to show what nonsense mathematics is, because by arguing in different ways about the same problem, one can come to different conclusions. We have presented an argument of our own without showing what is wrong with his. So how shall we address this? By pointing out that he hasn't done his conditioning properly. If all we knew was that one door or another was not the door with the prize behind it, then, yes, having been given that scrap of information and that scrap of information alone, we would indeed be left with a 50% conditional probability of each of the remaining doors being the one with the prize behind it. But the way the game is played, a little extra information manages to slip through, distorting those conditional probabilities.

Let us consider these random variables:



  1. W = the number on the door the prize is behind (1,2 or 3)
  2. C = the number on the door the contestant first chooses
  3. M = the number on the door Monty Hall lets the contestant look behind


We may reasonably take P to be uniformly distributed, with probability

p(W=1) = p(W=2) = p(W=3) = 1/3


since the show's producers would be foolish to choose any other distribution - an astute contestant could make use of that bias in the probabilities to increase his chance of victory. While C and W are statistically independent of each other, the contestant and the show's producers making their choices in a state of ignorance regarding the choice of the other, M can not be independent of either W or C, because it can't match either - Monty will not show the contestant what is behind the door that she has chosen, nor will he show her what is behind the winning door, regardless of whether or not the two are one and the same, which (of course) they might be. What will M be? If W does not equal C, Monty has only one choice: he has to show whichever door is left over. If W=C, we'll assume that he flips a mental coin, selecting either door with equal probability.

Has the reader noticed this, yet: if the contestant chooses door one and Monty shows her door two, while that outcome was a dead-on certainty if it was the case that she had chosen poorly (ie. the prize was behind door three), it was only 50% likely to occur if she had chosen well, and the same argument would apply if Monty had shown her the other door; in fact, it applies for any combination of choices. The situation doesn't look so symmetrical any more, does it? Oh, and we notice this:


.5 / (1 + .5) = 1/3


1/3 being the probability suggested by Vos Savant's argument above. Is this meaningful? Yes it is, as we shall see when we apply Bayes' theorem. Let us apply that theorem, from the point of view of a contestant who has chosen door one and has just been shown a goat pulling a cart behind door two:



P ( W = 1 | M = 2 ) = P ( M = 2 | W = 1 ) * P ( W = 1) / D

where D = ( P ( M = 2 | W = 1 ) * P ( W = 1 ) )

+ (P ( M = 2 | W = 2 ) * P ( W = 2) )

+ ( P ( M = 2 | W = 3) * P ( W = 3) )


As we already know, M can not equal W : that would correspond to Monty showing the door with the prize behind it. Thus ( ( M = 2 | W = 2) = 0, and the second term in D drops out, leaving us with



D = (1/2)*(1/3) + 0 * ( 1 /3) + 1*(1/3) = (1/2) * ( 1/3) + 1 * (1/3)


(The first condition probability - 1 /2 - being that resulting from Monry's mental coin toss when he has a choice of doors to show, because the contestant's first choice is right, and the third conditional probability, 1, representing the obligatory choice Monty is left with when the contestant first choise isn't the right door, and so there are two doors that Monty can't show). At this point, we notice that the factor of 1/3 drops out of the numerator and denominator, leaving us with 






                   (1/2) * (1/3 )                
            _____________________________ 
    
             (1/2) * ( 1/3)  + 1 * ( 1/3) 
   



                                   1 / 2 
                       =   _____________________ 

                               1 / 2   +  1 



                              1 / 2
                       =   ___________  =   1 / 3

                              3 / 2



bringing us back to that very suggestive piece of arithmetic we noticed earlier. Indeed, as a more careful application of our common sense would suggest, the conditional probability of the initial guess being right didn't go up when Monty showed that door. The conditional probabilities, however, still have to add up to 1, and the conditional probability of W = 2 dropped to zero the moment that Monty opened that door, and there's nowhere else to make up the difference, except for the value of P ( W = 3 | M = 2). What is Monty showed door three instead of door two? Almost the exact same calculation follows, and likewise in cases in which the contestant startd by choosing a different door than the first one. There is, then, no inconsistency in the calculations.

Some have noticed that the most excitable contestants seemed to win more often, leading to some cynical suggestions that the game was fixed for the sake of good ratings. In fact, the way the game is designed, if it is played by somebody who hasn't thought about what she is doing, excitability will prove a boon to the contestant, because it will incline her to do exactly what she should do - switch, increasing her chances of success to double of those of the more calm and collected (if statistically naive) contestant, who might feel that since Monty hadn't shown her anything that she didn't expect to see, that she had no reason to be feel jumpy enough to change her mind.




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