“In this world nothing is certain but death and taxes”
- Benjamin Franklin

Can you see the difference between these two statements:

“The Probability that a woman of age 40 has breast cancer is about 1 percent. If she has breast cancer, the probability that she tests positive on a screening mammogram is 90 percent. If she does not have breast cancer, the probability that she nevertheless tests positive is 9 percent. What are the chances that a woman who tests positive actually has breast cancer?”

(Most people think that the chance is 90%..)

and

“Think of 100 women. One has breast cancer, and she will probably test positive. Of the 99 who do not have breast cancer, 9 will also test positive. Thus, a total of 10 women will test positive. How many of those who test positive actually have breast cancer?”

Unlike the first statement, which is very confusing and a bit hard to grasp, the second is very easy and we can see that 1 in 10 women will test positive and not 9 in 10 as the first one might look to suggest.

This is uncertainty, and innumeracy, at its best. It clearly shows how easily fooled we can be just by a differently phrased statement! Gerd Gigerenzer’s book Reckoning With Risk brings up the horrible mistakes made by people using statistics and probability.

He shows how a lot of women have been lead to believe that they have breast cancer, when the chances are small and the chances that the tests are wrong are quite large – he also shows how physicians are ruled rather by their wallet than by what’s best for their patients; the costs of cancer screening is thus way larger than the benefits. In fact, there is no evidence that cancer treatment reduces death or is even remotely healthy for the patients. The errors the so-called experts make go into prostate cancer, AIDS, DNA, and various “facts” and statistics used to convict criminals (a lot of people are found guilty by the use of what is called “prosecutor’s fallacy”).

However, Gigerenzer shows how to turn this innumeracy into insight. He also shows how statistics can be used by public policy makers to manipulate outcomes into illusions of certainty. For example, there were, when the book was written in 2003, over 1,700 “trade organisations” promoting anything from asbestos to zinc, spending around $1 billion on “image advertisement” (the number has most likely increased since then). Consider the tobacco companies during this century and their promotion of smoking as not being negative to health but in fact positive to your health (all those doctors and authorities promoting smoking in commercials and ads)!

Lung cancer research, in the 20s and 30s, that showed the connection between smoking and cancer were dismissed because the researches that conducted the research were German, and in that era Nazis! (I will lay a claim that such modern forms of witch-hunting is still going on. Consider any thoughts or research done by people currently not acceptable, or talking about acceptable things are ostracized because of it – a more concrete example is the way fanatical nationalists claim that anyone who criticize or question the way the nation behave itself are traitors or hate their nation!).

Anther interesting social phenomenon in the book is: “Thesus monkeys reared in a laboratory, for instance, show no fear of venomous snakes. However, when a youngster watches an adult exhibiting fear of a snake, the youngster typically acquires this fear just by observing it once.”

Some things that Gigerenzer says can counter the way we read statistics is by using “Bayes’s rule” and “Natural frequencies”. I am very interested in games that train these ways of looking at statistics and probabilities. I am also interested in any games that play with the ideas of probability and statistics (such as the “Monty Hall Problem”, or “Three Prisoners Problem”) – if anyone has any such games, please, do share!

How do we come up with our worldviews? Why do we narrate the world in the ways we do? How are our many beliefs knit together? I have had time to think about this topic in several (really really really long) comments over at LDS & Evangelical conversations. I would advise reading through all of the comments in that thread when you get a few hours free.

I think that our worldviews, narratives, frameworks of belief systems represent what makes sense to us. These are the things we are inclined to believe, things that speak to our minds. Our narratives are things that bring us joy, peace, satisfaction, and accord with authenticity. I am sad that I have not gotten a chance to read Joseph Campbell in depth yet (seeing as I’ve brushed into him once before on the site), because I feel I am in complete accord with what he has said concerning “following your bliss.”

If you follow your bliss, you put yourself on a kind of track that has been there all the while, waiting for you, and the life that you ought to be living is the one you are living. Wherever you are—if you are following your bliss, you are enjoying that refreshment, that life within you, all the time.

The issue is in trying to figure out what “bliss” is, trying to figure out what our bliss is, and then trying to figure out what relation that has with truth.

I suppose I have been put off by this kind of idea. It is often peddled around sites that look kinda new agey *or* Eastern (I guess that’s the opposite: really old agey), and I’m not all that into those packages. So I am conflicted about linking to a site like this, which on the one hand seems to have applicable stuff, but which on the other hand seems to be a packaging of everything fuzzy about the new age (I already feel iffy about using the very term bliss, but oh well.)

But I think these ideas relate well with mine regarding authenticity.

The worldviews we use represent worldviews that make sense of things to us. The narratives we tells represent the story that best makes sense to us. We believe when we are personally persuaded or convinced to believe.

Authenticity is when we are faithful to things that “make sense” to us. It is a weighing of all things we are faced with — ourselves and our families, friends, and associates. All the possible actions we could take…the environment and our reactions to the environment. And at the end, we must assess and evaluate to what extent we value each of these things. When we have conflicts, we conduct cost/benefit analyses.When we are authentic, we are satisfied with our part in the choices we’ve made, and we feel we are on *our* track.

Inauthenticity is a failure to live in such a way. What does inauthenticity do to us? It brings misery…it brings a sense of internal death…that we are killing ourselves at a deep level. It is because we sense what we were meant to be or do, and when we are inauthentic, we do something else, or try to be someone else. This action is an attempt to lie to ourselves about who we are…but we cannot lie to ourselves. So instead, we fully experience that we are trying to stifle and suffocate ourselves.

I feel I have to address terms, however. I think everyone goes through this calculus, and everyone seeks authenticity. Everyone seeks “satisfaction,” “bliss,” “authenticity,” “peace,” “joy,” or whatever. But what are these terms? I think when people misinterpret the terms, they inadvertently become inauthentic to themselves.

Satisfaction isn’t simple happiness. It isn’t day-to-day thrill. It isn’t the absence of sadness. It isn’t what people would normally think of as being hedonism or related to it. It isn’t necessarily easy. It doesn’t mean a life without challenge or struggle. You can be sad, but satisfied. Alternatively, you can be happy, but miserable (from inauthenticity). You can (and most likely will) face great challenges and struggles in day-to-day living, but have peace through it all.

Satisfaction is something deeper. It is when things in a person’s life click together. It is when a person has found out a way to make sense. Ah, so 2+2=4 and I finally grasp it!

But this gets us to the final question. What is the relationship with truth?

Well…unless mathematicians do something incredibly scary with mathematical theory, we say that math is true. It is correct. It is objective and absolute and ultimate.

Most things in our lives do not have this solid backing, though. See…when we go by “what makes sense to us,” this is a subjective call. It depends on the *us* part more than any objective sense-making. As a result, it could be that what makes sense to us (which will satisfy us) could be incorrect. (Like if, say, “2+2=5″ made sense to us.) Consider that in math there are a great many unintuitive truths. Take the Monty Hall problem. Most people (even many mathematicians) intuitively believe the answer to be 50%. This satisfies them and makes sense to them, despite objective incorrectness. Hearing the answer of 2/3s is jarring and disturbing…fortunately, math has proofs (unless, as I said before, mathematicians do something incredibly scary with mathematical theory, putting everything we once knew into doubt) through which one can try to make sense.

Yet we don’t necessarily have these proofs in other fields and even when we do, we may not know it. So instead, what we believe to be proofs is confused by the curious phenomenon that we believe the hypotheses in question to be proofs because they make sense to us…when this is subjective to begin with.

This does not mean that truth does not exist. However, if we are going by what makes sense to us, wandering without “proofs” to show us when we are wrong in unintuitive situations, then will the truth even matter to us until a critical point in the future? How can this truth have more impact on us than authenticity, which will bring us bliss (or misery, if we do not adhere) every single day of our lives?

Math can be really fun. Seriously.

This post is the 2nd in a series of posts I’m planning to have about why math is such a beautiful, useful, and awe-inspiring subject, and that a lot of us can do math (advanced/seemingly difficult math even). Math is such an integral part of humanity since our cave dwelling days, and much more so now in most of our technology driven lives. Previously I wrote about how even advanced math, particularly advanced geometry, can be easily tackled with just your imagination. This time it’s about probability. I can just imagine some of you cringe at the thought of math, let alone probability. But I’ll try to show you that often times, logical reasoning is all that it takes to wrap your head around probabilities, even the ones that confound a lot of brilliant people, even some mathematicians themselves. In fact, we’ll end this article with a simulation of a game/game show. Not bad huh?

Probability and people

In a nutshell, probability is the area of math which deals with the likelihood of an event happening. It is usually expressed as a number, whether a fraction or a decimal, between 0 and 1, with a probability of 1 meaning the event will surely happen and a probability of 0 meaning the event won’t happen.

Now, don’t be too hard on yourself thinking that probability is too hard for you, unlike most of the human population. In fact, probability is one really confounding area of math and problems in it that seem to be easy in hindsight, turn out to be deceptively difficult or tricky, even for  mathematicians, teachers, and other brilliant men and women around the globe. In fact a lot of us have trouble wrapping our heads around probabilities. You mix that with human hopefulness and also the difficulty of grasping very large numbers and what you get is the staggering number of people around the world falling in line to get their lotto tickets so they could win the multi-million prize money.

In fact, if we do the math, in a typical 6/49 game of lotto (6 unique numbers chosen out of 49 numbers, where the order of the 6 numbers is not important) we find that your chances of winning today after buying that lotto ticket is 1 in about 14,000,000. So if Lucy (one of the earliest hominids/proto-humans known to us) or her people, or perhaps even Neanderthals started betting on the lottery at the beginning of their lives, some of them should be millionaires by now. That’s how bad we are at assessing odds, especially coupled with large numbers. So when you go buy that lotto ticket later, I’m afraid the odds are so much against you.

However, I’ll discuss next a particularly perplexing probability problem pondered by people, even brilliant ones, and found the solution to be deceptively trivial after all. Actually, even after you get the explanation, from a practical standpoint it doesn’t seem like so. But the logical reasoning will quite surely buy you out. But don’t fret, all you need again is imagination and logical reasoning.

Game time

Some of you may have heard/read about the American game show Let’s Make a Deal. The Monty Hall problem (MHP) was named after the show’s host. Simply stated, the rules of the game are as follows:

The game master (GM), has 3 doors: 2 with goats behind them and one with a car behind it. The GM lets you choose one door, which you think holds the prize car behind it. Since the GM’s job is to make you and the audience excited and enjoy the game, the GM opens another door. But since the GM knows the placement of the goats and the car i.e. which door has which item behind it, the GM opens a door which has a goat behind it. Now, the GM poses a question to you: Do you or do you not want to change the door you initially picked i.e. the GM gives you an opportunity to stay with the door you originally picked, or to choose the other door, knowing that one of the doors, which the GM opened, has a goat behind it.

The GM in the show is of course Monty Hall (MH). Now, you’d most probably think that since there are only 2 doors left unopened, that the probability of getting either a goat or a car is now 50/50 or 50% right i.e. it doesn’t matter whether you switch doors or not?

Nope.

In fact, however counterintuitive this may seem, your chances of getting the car at this point of the game doubles if you decide to change the door you initially picked. How? Let’s find out shall we?

Goat, Car, Goat

Now let’s strap on our imagination and logical reasoning caps to find out how the probability of getting the car increases two-fold if you switch your chosen door, and that it’s not a 50/50 chance of getting the car once a door with a goat has been opened by the GM.

Monty Hall problem

One way of looking at how this counterintuitive probability problem is correctly tackled is by taking the possibility of the events one at a time (refer to the figure above please). In this scenario we show that when you switch doors, you always double your chances of winning. Here’s how:

1. First event, say you picked a door and it happened to have the prize car behind it.  Regardless of which door the GM opens, switching in this case either gives you goat A or goat B i.e. you lose the prize car. Out of the 3 possible scenarios (2 of which are listed right after this one), in this one event/case do you lose the prize car.

2. Second event, you choose a door with a goat (goat A) behind it. The GM opens a door again with a goat (goat B) behind it. If you switch in this case, you get the car. This event, wherein you get the car by switching, is one event which you get the prize car. Score one for you.

3. Third event, you choose the 3rd door with a goat(goat B) behind it. The GM again opens a door with a goat (this time, goat A). So when you switch, you get the car again. Yay. This event, wherein you again get the car by switching, is another event which lets you take home the prize.

So what did we get from all this? We saw that out of 3 events/cases of picking either of the 3 doors, you always get 2 events (event 2. and 3.) which favor switching and which lets you walk away with the prize (or in this case, drive away with the prize). So the odds of getting the car/prize in the MHP is not 50% as a lot of us would initially assume, but instead, is really 2/3 or approximately 66.7%.

It can take a while to sink in, but the reasoning/explanation is quite logical and sound.

Try it out!

I actually tried this out with my mother and at another time with my younger brother. What I did was I got 3 opaque plastic cups (simulating the doors) and 2 toy cows (no goat toys in our house at that time) and 1 robot toy that transforms into a car (not bad for a prize no?). I made them act as a GM at one time, with me being the game contestant. Of course to prove my point I always switched. We did this about 20 times and I got the prize car (or robot) at around 14 times out of the 20 (roughly 2/3 of 20). Then I acted as a GM and they acted as the contestant. Then their job was not to switch doors (or cups), just to prove my point that you get the prize more often than not (2/3 of the time remember?) by switching instead of staying with your original door/cup.

They even asked me if I was doing a magic trick on them. I told them it was the power of mathematics and of logical thinking. Imagine what much more primitive, let’s say Bronze-aged men, would think of me, with this knowledge, even without modern devices like a cellphone. Perhaps they’d think of me as an oracle or even a god.

Great, great. But what’s the use?

I think one important thing we can get from this (other than to show you that you can do maths you thought were too hard or complicated for you) is that with math, we can make decisions in our lives (sports betting, lottery, game shows and so on) with more clarity, logic, and sound reasoning, instead of just blind optimism.

If you didn’t get the logic on how to win the game at first glance, or if you thought it was 50/50, don’t be ashamed, a lot of people (some brilliant even) fell for it too. In fact, out of 228 subjects in a study, only 13% chose to switch, and that the rest (87%) assumed that the switching didn’t matter since the likelihood of getting the car out of the 2 unopened doors are equal (research by Mueser and Granberg, 1999).

Quoting cognitive psychologist Massimo Piattelli-Palmarini

“… no other statistical puzzle comes so close to fooling all the people all the time”

and

“that even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”

So, not bad eh? Still think math (or at least those areas you think are too advanced or complicated for you) isn’t for the average person? If so, then look forward to my next posts about math.

References, resources, and further reading

• Wikipedia article on the mathematics of lottery
• Wikipedia article on the Monty Hall problem

When I was younger, I would often play a game with myself: He knows.  I know he knows.  He knows I know he knows. I pondered these sentences, turning them over slowly in my mind, thinking that eventually, I would have a feel for them.  Unsurprisingly, for the mind of a little boy, the examples that came to mind were adversial.  He tried to out know me, as I tried to out know him.  So, I could see this as a game of countering strategies.  When I felt I had an intuition built up, I tried for more,

I know he knows I know he knows.

He knows I know he knows I know he knows!

I know he knows I know he knows I know he knows!!

He knows I know he knows I know he knows I know he knows!!!

I know he knows I know he knows I know he knows I know he knows!!!!

There was always a point at which the sentences attained true meaninglessness.  It was at that point, I gave things a rest until next time.

Today, I share this game for reason.  I realized that in two mathematics puzzles, the real break through comes from getting inside the other guy’s head.

There is a problem called the Monty Hall Problem which most people seem to get wrong.  It comes in the form of a simple game, with a simple goal and a simple dilemma.  One is presented with three doors by the game show host: behind one of the doors is a prize and the goal is to pick the door behind which the prize lurks.  One has a first opportunity to pick a door and having picked one, another door, different from the one you picked is opened.  Subsequently, one is given a choice of switching the door to the remaining yet unopened door or sticking with the door that one has already picked.  The central question is this: is it a better strategy to stay with the original choice or to switch doors?

Most people get this wrong.

Most people think it doesn’t matter either way.

This is wrong.

Our intuition does not seem to work in this situation.  SO WHAT’S GOING ON HERE?

It’s a matter of perspective.  Let me explain — with diagrams.  From the perspective of the participant in the game,  this diagram lays out the scenario.   First, one chooses a door.  Second, the gameshow host makes a choice of another door.  Finally, the participant makes a choice between the remaining doors.   (One should note that the diagram only shows what happens when door one is chosen first.   The name of the door doesn’t matter.   For instance, before we play the game, we can always have an assistant quickly rename all the doors so that the first door we choose is always  called door number one.  As long as we don’t move around the labels during the rest of the game, it does not change the reasoning.)

The next two diagrams highlight that in the case of switching and in the case of staying with the same door, there are two situations where one can win and two situations where one can loose.

What’s the catch?  The key insight is looking at things from the perspective of the host and realizing these pictures are not the whole story.   The host made one more move even before the participant makes the first: the host chooses where to place the prize.  As the first two diagrams show, the host is playing a secret game of avoiding the prize.   If  any of the first two situations occurs, the host telegraphs where the prize is by avoiding it.  Therefore, by switching one can take advantage of this weakness.

In the last case, the participant has guessed correctly.

Switching is a losing move here.  In the first two cases, switching is always a good idea.   So there  are two cases where switching is always a win and one case where switching loses the game.   Therefore, switching is the better strategy which wins two out of three times.   This is the Monty Hall Problem.

Although the common failure of intuition in this problem is often cast as a mathematical failure or a failure of knowledge of probability, I think that the issue is not realizing when the game started, which is before the participant ever makes a move.

If you would like to try some of these ideas for yourself, I have provided programs in three forms:

HERE

There is an executable file which should run if you double click it on any windows system. There is a Mathematica Notebook; and there is the C++ source code for the executable.

The Monty Hall Paradox

One of the 3 doors hides a car. The other two hide a goat each. In search of a new car, the player picks a door, say 1. The game host then opens one of the other doors, say 3, to reveal a goat and offers to let the player pick door 2 instead of door 1. Is there an advantage if the the player decides to switch? (Courtesy: Wikipedia)

Hola amigos! Yes, I’m back! It’s been eons and I’m sure many of you may have been wondering why I was MIA. Let’s just say it was academia as usual.

This post is unique as it’s probably the first where I’ve actually learned something from contributors and feedback. A very critical audience and pure awesome discussion. The main thrust was going to be an analysis of the question, “If you had to pick an answer in an MCQ randomly, does changing your answer alter the probabilities to success?” and it was my hope to use decision trees to attack the question. I first learned about decision trees and decision analysis in Dr. Harvey Motulsky’s great book, “Intuitive Biostatistics“. I do highly recommend his book. As I pondered over the question, I drew a decision tree that I extrapolated from his book. Thanks to initial feedback from BrownSandokan (my venerable computer scientist friend from yore ) and Dr. Motulsky himself, who was so kind as to write back to just a random reader, it turned out that my diagram was wrong and so was the original analysis. The problem with the original tree (that I’m going to maintain for other readers to see and reflect on here) was that the tree in the book is specifically for a math (or rather logic) problem called the Monty Hall Paradox. You can read more about it here. As you can see, the Monty Hall Paradox is a special kind of unequal conditional probability problem, in which knowing something for sure, influences the probabilities of your guesstimates. It’s a very interesting problem, and has bewildered thousands of people, me included. When it was originally circulated in a popular magazine,  “nearly 1000 PhDs” (cf. Wikipedia) wrote back to say that the solution put forth was wrong, prompting numerous psychoanalytical studies to understand human behavior. A decision tree for such a problem is conceptually different from a decision tree for our question and so my original analysis was incorrect.

So what the heck are decision trees anyway? They are basically conceptual tools that help you make the right decisions given a couple of known probabilities. You draw a line to represent a decision, and explicitly label it with a corresponding probability. To find the final probability for a number of decisions (or lines) in sequence, you multiply or add their individual probabilities. It takes skill and a critical mind to build a correct tree, as I learned. But once you have a tree in front of you, its easier to see the whole picture.

Let’s just ignore decision trees completely for the moment and think in the usual sense. How good an idea is it to change an answer on an MCQ exam such as the USMLE? The Kaplan lecture notes will tell you that your chances of being correct are better off if you don’t. Let’s analyze this. If every question has 1 correct option and 4 incorrect options (the total number of options being 5), then any single try on a random choice gives you a probability of 20% for the correct choice and 80% for the incorrect choice. The odds are higher that on any given attempt, you’ll get the answer wrong. If your choice was correct the first time, it still doesn’t change these basic odds. You are still likely to pick the incorrect choice 80% of the time. Borrowing from the concept of “regression towards the mean” (repeated measurements of something, yield values closer to said thing’s mean), we can apply the same reasoning to this problem. Since the outcomes in question are categorical (binomial to be exact), the measure of central tendency used is the Mode (defined as the most commonly or frequently occurring thing in a series). In a categorical series – cat, dog, dog, dog, cat – the mode is ‘dog’. Since the Mode in this case happens to be the category “incorrect”, if you pick a random answer and repeat this multiple times, you are more likely to pick an incorrect answer! See, it all make sense ! It’s not voodoo after all !

Coming back to decision analysis, just as there’s a way to prove the solution to the Monty Hall Paradox using decision trees, there’s also a way to prove our point on the MCQ problem using decision trees. While I study to polish my understanding of decision trees, building them for either of these problems will be a work in progress. And when I’ve figured it all out, I’ll put them up here. A decision tree for the Monty Hall Paradox can be accessed here.

To end this post, I’m going to complicate our main question a little bit and leave it out in the void. What if on your initial attempt you have no idea which of the answers is correct or incorrect but on your second attempt, your mind suddenly focuses on a structure flaw in one or more of the options? Assuming that an option with a structure flaw can’t be correct, wouldn’t this be akin to Monty showing the goat? One possible structure flaw, could be an option that doesn’t make grammatical sense when combined with the stem of the question. Does that mean you should switch? Leave your comments below!

Hope you’ve found this post interesting. Adios for now!

Copyright © Firas MR. All rights reserved.

Readability grades for this post:

Flesch reading ease score:  72.4
Automated readability index: 7.8
Flesch-Kincaid grade level: 7.3
Coleman-Liau index: 8.5
Gunning fog index: 11.4
SMOG index: 10.7

Readings:

Intuitive Biostatistics, by Harvey Motulsky

The Monty Hall Problem: The Remarkable Story Of Math’s Most Contentious Brain Teaser, by Jason Rosenhouse

usmle, decisionanalysis, decisiontree, examination, examinationtechnique, “examination technique”, “decision analysis”, “decision tree’, montyhallproblem, “monty hall problem”

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I had heard about this movie with relation to Mathematics. I have watched A beautiful Mind-the life of a mathematician, earlier, so I was waiting to see this movie. Yesterday I could watch it on TV-courtesy HBO. The script writers, and director Robert Luketic have taken care that all the technical stuff (including Mathematical theorems) are portrayed correctly, unlike some of Bollywood movies – Ajnabiee (password is shown in plain text), Fida (Shahid Kapoor hacks by typing in “c:\hack.exe” in cmd), etc wherein technical stuff shown has no meaning at all. Anyway, instead of going deeper into how not to make a movie (Archive, moviemaker), I am going to tell the story of movie here.

MIT senior math major Ben Campbell is accepted into Harvard Medical School but cannot afford the $300,000 cost. Despite boasting a 44 MCAT score and a 4.0 GPA, Ben faces fierce competition for the prestigious Robinson Scholarship which would pay for medical school. He is told that he needs a way of “dazzling” Harvard in some way to stand out from from the other extremely well-qualified applicants (a life experience).

Professor Micky Rosa challenges Campbell with the Monty Hall problem (this is one of the probability theory paradoxes. In a nutshell, Monty Hall problem can be stated as- 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’s 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? ), which Campbell solves successfully. Rosa invites Campbell to join his blackjack team, which consists of fellow students Choi, Fisher, Jill and Kianna. The system involves card counting, and the team—shepherded by Rosa—is split into two groups. “Spotters” play the minimum bet and keep track of the count. They send secret signals to the “big players,” who place large bets whenever the count at a table is favorable. Campbell reluctantly joins the team, telling Rosa he is only doing so until he can pay for medical school.

Rosa takes the team to Las Vegas over many weekends; Campbell comes to enjoy his luxurious lifestyle there. His performance impresses Jill (who falls for him) and Rosa, but Fisher becomes jealous at Campbell’s blackjack success. Rosa kicks a drunken Fisher off the team after he insults Campbell and incites a melee that requires the team to quickly “cash out” (using dancers from their usual strip-club meeting place) before the casino switches chips. Meanwhile, security chief Cole Williams monitors the blackjack team, particularly Campbell.

Campbell, distracted by blackjack, botches his part of a project for the 2.09 engineering competition, estranging him from his pre-blackjack friends. During the next trip to Vegas, an emotionally-distracted Campbell continues playing even after he is signaled to walk away, losing $200,000. An angry Rosa leaves the team and demands Campbell repay him for the loss. Campbell and his three remaining teammates agree to go into business for themselves. Williams apprehends Campbell, physically assails him, then lets him go after giving him a death threat.

Upon his return to Boston, Campbell learns that he has been given an incomplete for one of his classes and therefore will not graduate, and that his winnings have been stolen from his dorm room. He suspects that Rosa is behind everything but has no evidence. Campbell reconciles with his friends and Jill, and approaches Rosa with an offer: He and the team will hit Vegas for one more attempt before the casinos install biometric software that will quickly identify card counters, as long as Rosa—himself once a very successful “big player”—also plays.

Disguised, the team returns to the Planet Hollywood and win $640,000 before fleeing with their chips from Williams and his men. Campbell and Rosa split up, with Rosa taking the bag of chips. Rosa escapes with the intention of stealing the winnings, but finds his bag is full of chocolate coins and his limo is being driven by the casino manager.

The audience then learns that Williams had made a deal with Campbell after beating him up; he would let Campbell come to Vegas for one last night to make a lot of money in exchange for Rosa, who years earlier cost Williams a casino job by winning a seven-figure take via counting cards. Campbell’s pre-blackjack friends joined the team to help their friend. After capturing Rosa, Williams confronts Campbell and double-crosses him by demanding at gunpoint the bag of chips for his retirement. Aware that Ben plans on attending medical school to be a doctor, he assures the young man that everything will work out for him in the end. Ben hands the money over to Williams and leaves. Moments later, Rosa is tied to a chair where Williams greets him, informing the professor that he will turn him over to the IRS for evading taxes on his winnings. Campbell loses money The movie closes with Campbell recounting the entire tale to a “dazzled” Harvard administrator.

Kevin Spacey (as Professor Micky Rosa) tried to imitate University Profs and I must say he was quite successful. The way he portrayed the character, reminded me of Richard Feynman, whose book ‘You are surely joking Mr. Feynman’ I am currently reading and whose lectures on Physics are available online for free.

There is a famous puzzle known as the ‘Monty Hall Paradox’. 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’s 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?

Yes, by choosing again you actually improve your chances of winning.

I like this challenge because it is counter-intuitive. Without applying mathematics, I would not know that trusting my instincts can be counter-productive, even dangerous. Being aware of this, I wonder how often my choices are poor, when I could be using thinking skills to improve my chances for success.

While Bnei Yisrael are in the desert, they repeatedly make poor choices. They complain again and again about food and water. Even the leaders err when they are sent to spy on the land of Israel and are punished for their report which causes the people to complain about entering the land. If I was in their shoes, wandering through the desert, I also imagine that my intuition would result with similar complaints.

Complaining may be a Jewish trait that has its roots with Bnei Yisrael in the desert, but we see how little patience God has for such complaints. When I read these stories with 20-20 hindsight, I also find their complaints ridiculous. Yet, I recognize that most of my decisions are also based on gut feelings, so how do ensure that I make good decisions?

Our challenge is to know when to listen to our own inner voices, especially when it leads to shouting out and advocating for social justice. After all, a revolution is good every once and a while. But we must remember that sometimes truth, or what is most correct, is counter-intuitive. How do we do remember this? One way is to be self-reflective. We can use tools such as the sciences or friends and family to keep ourselves in check.

I sense that by knowing when and how to be counter-intuitive we can improve our chances at becoming millionaires, or at least becoming better people. Which counter-intuitive practice do you recommend for me?

200th post y’all!!

1. I cut a freaken huge paper cut…………… and it hurt twice as much as the time I cut myself with a bread knife.
2. 98.1 Math (i hate myself)
   98.3 – French (um no comment)
   96.8 – Science (woooot)

   as you can see my marks have dropped an avg of 1-3%. Why oh why?

3. PE class ran all the way to tim horton’s @ fraser highway because today was free “iced latte” day. The run there was brutal due to the horribly hot weather. When I got to tim’s I was drenched with sweat it was so DAMN hot outside. I saw 2 coworkers there and I think they’re dating hahahaaha. They saw me right when I looked/felt like a thousand bucks. The plus side is we got to walk back.
4. I want to learn how to count cards(for black jack, etc.) Must watch movie “21″
5. I don’t understand the montyhall problem. I’ve seen/witnessed it 4 times already but I have yet to understand why changing our answer before the end gives you a 2/3 chance. It should be 1/2!!!!!
6. Conditional probability makes no FREAKEN SENSE….well some of it does. The other half i’m just trying to grasp.
7. yay my book came and I’m going to pick it up tomorrow yayayayay.
8. why can’t I see angela’s blog? I want to know what’s going on in her life!!!!!!

Imagine you are a contestant on the 70’s game show Let’s Make A Deal with host Monty Hall. You’ve made it to the final round, and have a chance to win a brand new car!

Monty asks you if you want what is behind Door #1, Door #2 or Door #3. Before you choose, Monty explains that there is a new car behind one of the doors, and a goat behind each of the other two doors.

You select Door #1. Before revealing your prize, Monty opens Door #3 and reveals a goat. The new car is behind Door #1 or Door #2! Then Monty asks you if you want to change your selection from Door #1 to Door #2.

Assuming you would prefer to win the new car instead of a goat, is it to your advantage to change your selection?

You might be surprised to learn that the answer is yes! While at first it might appear as a 50/50 proposition, it is not.

When you choose Door #1, you had a 1 in 3 chance of being correct, or inversely, you would pick a goat 2 out of 3 times. After Monty opens Door #3 to reveal one of the goats, you still have a 1 in 3 chance of winning a new car if you retain your original selection of Door #1. In other words, your original choice has a 2 out of 3 chance of being incorrect.

If you change your choice to Door #2, you have the inverse of keeping your original selection, or a 2 out of 3 chance of being correct! You will win the new car 2/3rds of the time by changing your choice in this example.

This is an example of what is commonly referred to as the Monty Hall Problem,  which according to Wikipedia is a probability puzzle with a counterintuitive solution. It’s the mathematical equivalent to the Three Prisoners Problem and the Bertrand’s Box Paradox.

This is something that has been around for quite a while, but I just discoved it after reading an article in Card Player Magazine which I subscribe to (I enjoy playing poker, especially No Limit Texas Hold’em and Omaha).

Here is a webpage where you can play the Monty Hall game yourself, and track your own results.

Monty Hall Problem kısaca şöyle;
Bir televizyon şovundasınız, önünüzde 3 kapı var, bir tanesinin arkasında araba, diğer ikisini arkasında keçi var. Bir kapı seçeceksiniz ve arkasındaki ödülün sahibi olacaksınız. Kapıların arkasında hangi ödüller olduğunu bilen sunucu size bir kapı seçmenizi söylüyor. Bir kapı seçiyorsunuz. Ama o kapı açılmadan önce sunucu sizin seçmediğiniz diğer iki kapıdan arkasında keçi olanını açıyor ve size seçtiğiniz kapıyı değiştirme şansı veriyor. Burdaki sorun ilk seçtiğiniz kapıyı değiştirmeli misiniz, seçtiğiniz kapıda ısrarcı mı olmalısınız, yoksa değiştirip değiştirmemeniz kazanma şansınızı etkilemez mi?

İlk bakışta siz bir kapı seçtikten ve sunucu diğer kapılardan arkasında keçi olanı açtıktan sonra seçtiğiniz kapıyı değiştirseniz de değiştirmeseniz de arabayı kazanma şansınız %50 gibi görünüyor ( Yani değiştirmek kazanma şansınızı etkilemiyor). Ama biraz daha dikkatli incelersek kapıyı değiştirmenin bize daha fazla kazanma şansı verdiğini görebiliriz.

İki seçeneğimiz var, ya teklif geldiğinde kapımızı değiştireceğiz, ya da ilk seçtiğimiz kapıda ısrarcı olacağız.

ilk seçtiğimiz kapıda ısrarcı olmak ( değiştirmemek ):
İlk başta 3 kapı varken arabayı bulma şansımız %33 ve biz nasıl bir teklifle karşılaşsak da kapımızı değiştirmeyeceğimize göre sonuçta da arabayı bulma şansımız %33 olacaktır. Şöyle düşünün, teklif gelmesin, 3 kapıdan bir kapı seçin, direk açılsın kapılar. Arabayı kazanma şansınız %33 olacaktır. Nasıl olsa değiştirmeyeceksiniz.

Kapıyı değiştirmek:
Çok basit mantıkla, ilk başta %33′lik şans ile arabayı bulursanız ve teklif sırasında kapınızı değiştireceksiniz ve kaybedeceksiniz. İlk seçtiğiniz kapıda %67′lik şans ile keçi olan kapıyı seçerseniz, kapınızı değiştireceksiniz ve kazanacaksınız. Sonuç olarak %33  kaybedersiniz, %67 kazanırsınız.

Kapı sayısını 100′e çıkaralım, bir kapı seçtik, sunucu diğer 99 kapıdan 98′ini ( keçi olan ) açtı ve bize seçtiğimiz kapıyı değiştirip değiştirmeyeceğimizi sordu. Tabiki değiştirmeliyiz. Arabanın bizim seçtiğimiz kapıda olma şansı 1% iken diğer 99 kapıdan birinde olma şansı 99%’dur. 99 kapıdan 98′inde keçi olduğunu zaten biliyorduk, ama hangilerinde olduğunu bilmiyorduk. Sunucu bize yardım etmiş oldu.

Her zaman kapıyı değiştirelim ve rastgele üretilen kapılardan rastgele bir kapı seçelim. Bir parça c kodu ile kanıtlamaya çalıştım:

#include <stdio.h>
#include <stdlib.h>
#include<time.h>
#define COUNT 1000

int main()
{
 int kapi[3];
 int i,temp,win=0,lose=0;

 srand ( time(NULL) );

 for ( i=0;i<COUNT;i++ )
 {

  kapi[0]=0;
  kapi[1]=0;
  kapi[2]=0;
  temp=rand()%3;

  kapi[temp]=1;

  temp=rand()%3; 
  
  if( kapi[temp] == 1 )  // ilk seçişte kapıyı bulursak
  {
   lose++;
  }else
  {
   win++;
  }

 }
 printf("win = %d\nlose = %d\n",win,lose);

 return 0;

}

Sonuçlar:

win = 675 lose = 325
win = 667 lose = 333
win = 677 lose = 323
win = 672 lose = 328
win = 653 lose = 347
win = 693 lose = 307
win = 679 lose = 321
win = 681 lose = 319
win = 682 lose = 318
win = 672 lose = 328

Gördüğümüz gibi ilk seçtiğimiz kapıyı değiştirmek bize arabayı kazanmamızda daha fazla şans sunuyor. Böyle bir yarışmaya katılırsanız ilk seçtiğiniz kapıyı mutlaka değiştirin, tabi bir keçi kazanmak istemiyorsanız.

Monty Singh was a wise man. The brightest in the land.

On that eventful Thursday, he received an email. This was a mail  like.no.other.

It was from a Nigerian King.

A real friggin rich Nigger Raja. [P.C version -  Niger. But different country]

Monty Singh was a pyoor Veggie. He didn’t like spam.

But this mail had to be genuine. His IIT alumnian brain could sense it. It was authentic. Right down to the black fonted signature in Wingdings.

He glanced through the contents. He couldn’t believe it. He read it again, this time slowly, and only then did the weight of the matter dawn on him.

He was chosen to participate in a Game show. A quiz of sorts.

Monty smirked. He was an ace quizzer. How he missed those days.

Flashback : Brought to you by Chintu Candy.

It was in seventh standard. He had had his morning’s cuppa’ Horlicks.

Then he went to BQC, thrashed Derek O’Brien mostly left, and occasionally right as well. Pinky Singh was a proud mother that day.

Monty came back from his reverie. He had to think this through.

Monty loved Probability. He simulated a random bit generator. Lady Luck was with him. “Go to Nigeria, you worthless bastard!”, she bellowed.

The queue for the Visa was shorter than he had expected. There was just one local brown model visiting the country for a Fair and Lovely – Limited Nigerian Edition ad-shoot. He grinned as he saw the neighbouring ‘US of A’ Visa line, mostly consisting of bespectacled grad wannabees.

He was received in Nigeria, amidst a royal fanfare. He was led to the only 7 star hotel in the country -  Bobby Da Dhaba. Monty felt right at home.

He woke up that morning, and got himself a beer.

Oops. Wrong post!!!

Monty was up and soon spiffily dressed himself. His father’s pink tie would go well with his lemon yellow shirt.

Karan Johar, the host, looked surprisingly hetero that morning. Must be all that Koffee, thought Monty. “Never mind his temporary non-gayness”. “Concentrate”, he said to himself, as he walked to the stage, which was lit by a thousand colour-colour LED’s. A sight to behold.

Monty raced through the questions like Usain Bolt on steroids. They didn’t call him “Monty Mastermind” just like that.

The final question. This was a toughie. Monty kept his cool. He worked it out. Ruddy Brilliant. He was dingchakkingly good.

“And now Mr.Monty. How bout a bonus round”, shrieked Johar.

“ A flirty car, or you lose it all…..”

No, wait. No one had told him about a bloody bonus round.

As if reading his concerns, Johar replied, “ Don’t worry, Its just a tiny game of probability”.

Gosh. Monty almost had a tiny orgasm.

“ Very similar to the Monty Hall scenario, I take it that you know about it”, asked Johar.

“Pfft. Know about it? Why do you think my dad named me Monty?”

“Oh. I thought that was because you like to…..  Never mind…”

“Ok. All the doors are hidden behind this wall. Just for kicks”. “And…”

“Oh. Will you start already”. “I choose door no.2”. “Which car is it btw?”

“Premier Padmini’s hot friend, Diablo Lamborghini….” “ Whate joke . Whate joke. Ha . Ha.. I know . I can be a pain in the bottoms sometimes”, quipped Karan Johar.

“Ok. Mr.Monty. I’ll open door no.1 and… WTF…”.

“Damn you, Nigerians, stop touching my goat”.

Monty’s brain started working faster than a computer. All those nuggets from Dasgupta, and T.M.H, heck even some from Krishna’s came back to him in a rush. He evoked Bayes, and his conditional Probabilistic models. And in a jiffy, the answer was gambolling right in front of his eyes.

“So, Mr. Monty, what’s your call? Will you flip your choice, or keep it?”

“ Duh. Flip my choice. Obs”.

“Ok. Have it your way.” ……. “ “Hurrah, You win….”

…..

…..

…..

…..

“ the goat”.

“There were only two doors. Retard”….

All rise for the Nigerian Anthem.

P.S : Monty Singh was a wise man. The brightest in the land.

Update – 7-10-2008

Atul asked me whether this was a Himesh Reshammiya belting post? Actually, I am currently cursing myself for not noticing that Himesh is playing Monty’s role in Karzzzzzzz ( Did I miss a ‘z’? ).

Quoting Himesh – “ Rishi Kapoor is the best-looking Monty, I’m the worst” – We agree.

That, friends, is a different Full-Monty-Problem altogether.

Vor einiger Zeit, im Aufenthaltsraum des ETH-Webradio Radius, fragte mich ein Student: “Stell dir vor, du bist in einer Talk-Show. Du kannst zwischen drei Toren wählen, hinter einem befindet sich der Hauptpreis, hinter den anderen ein Zonk. Nachdem du gewählt hast, öffnet der Moderator eines der beiden anderen Tore, in dem sich ein Zonk befindet. Nun fragt er dich, ob du dein Tor wechseln möchtest. Was ist die richtige Entscheidung?”

Wie aus einem Pfeil schoss die richtige Antwort aus mir raus: “Kommt nicht drauf an!” Er und ein handvoll weiterer ETH-Stundenten aus den Bereichen Informatik, Neuroinformatik und Maschinenbau versuchten mich eines Besseren zu belehren: Wechseln sei ganz klar die bessere Lösung, das stände sogar im Internet. Ihre dürftigen Berechnungen will ich dem Leser ersparen, sie wären Anlass genug, der ETH Steuergelder zu entziehen.

Was die Studenten versucht haben zu formulieren, nennt sich Monty Hall Problem. Es impliziert, dass der Moderator immer ein Tor mit einem Zonk öffnet, und den Kandidaten fragt, ob er wechseln möchte. In diesem Fall steigt die Gewinnchance durch einen Wechsel von 1/3 auf 2/3. Bietet der Moderator aber nur einen Wechsel an, wenn der Kandidat das Tor mit dem Hauptpreis gewählt hat, sinken seine Gewinnchancen von 1/3 auf 0.

The solution to the Monty Hall problem (switching wins you 2/3 of a car) depends for its answer on the fact that you know how Monty will act. Other host behaviours are possible. So my question is this: what is the best strategy if you don’t know what Monty’s behaviour is? Is it different in single case vs long run scenarios? In the latter case, what about a strategy that allows you to alter your behaviour depending on Monty’s behaviour? I don’t really know how to answer these questions; I have enough trouble convincing myself of the solution to the original problem!

In other news, a couple of books by D.H. Mellor are available for free online! Matters of Metaphysics and The Matter of Chance. And more philosophy gubbins- Philosophy Bites: Bitesize philosophy podcasts. Wonderful.

One last thing. Tim Minchin and Duke Special look quite similar. They both play piano type music. But Tim Minchin is from Australia and does comedy songs and Mr. Special is from Northern Ireland and plays “proper music.”

Tim Minchin

Duke Special

Most every one has heard of Let’s Make a Deal. Perhaps not as many people have heard of Marilyn vos Savant, but she writes a column in Parade magazine that comes as an insert in many Sunday newspapers, called Ask Marilyn. She is famous for having been listed in Guinness World Records Hall of Fame for having the highest IQ.

OK, back in September 1990, there appeared this question,

“Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, ‘Do you want to switch to the other unopened door?’ Is it to the contestant’s advantage to make the switch?”

What’s the answer?

Be careful, tons of people, including many math Ph.D. holders got this wrong.

The problem is now referred to as the Monty Hall problem. And is given a prominent spot on her website.

Im Film 21, den ich neulich gesehen habe wurde ein mathematisches Problem erwähnt, welches mir zum ersten Mal auf dem Vortrag ‘Absurde Mathematik‘ von Anoushirvan Dehghani, der auf dem 24C3 gehalten wurde begegnet ist.

Es geht dabei um das sogenannte Ziegenproblem. Die Situation dabei ist folgende:

Ihr spielt ein Spiel. Dabei habt ihr die Chance ein Auto zu gewinnen. Das ist toll
Und so funktioniert das Spiel: Ihr steht vor drei Toren. Hinter zwei dieser drei Tore befindet sich eine Ziege. Hinter dem dritten aber ist euer Auto.

Nun sollt ihr eine Tür auswählen. Der Spielleiter öffnet dann eine andere Tür hinter der eine Ziege steht.

Er gibt euch dann die Chance erneut zu wählen. Entweder bleibt ihr bei eurer Tür oder ihr ändert eure Wahl.

Die Frage ist nun – was bringt euch mehr? Die Wahl zu ändern oder dabei zu bleiben?

Ich habe das ganze mal in einem Programm modelliert, welches diesen Versuch hunderttausendmal mit Wechsel und hunderttausendmal ohne Wechsel durchführt. (Man kann sich das Programm hier anschauen)

Das Ergebnis ist nun folgendes:

Häufigkeit ohne Wechsel: 32942
Häufigkeit mit Wechsel: 66922

Das heißt die Chance mit Wechsel zu gewinnen liegt bei circa 2/3, während die Chance ohne Wechsel bei 1/3 liegt.

Das verwirrt oder? Man würde doch eine gleiche Häufigkeit erwarten. Also – warum ist das so?

Schauen wir uns doch mal die Wahrscheinlichkeiten anhand eines Baumes an.
A, B und C sollen die Tore sein. Der Spieler wählt das Tor A (lila) aus. Der Spielleiter öffnet dann das Tor C hinter dem eine Ziege steht.

Alle drei Tore haben zunächst die gleiche Gewinnwahrscheinlichkeit. Sie ist 1/3. Wählt der Spieler nun ein Tor, so hat dieses die Wahrscheinlichkeit 1/3, die anderen Tore haben zusammengenommen die Wahrscheinlichkeit 2/3.

Nun wird eines der beiden anderen Tore als falsch gekennzeichnet (hier Tor C) und entfällt somit. Damit hat das eine Tor, welches noch übrig bleibt von den beiden anderen (hier als B gekennzeichnet) die Gewinnwahrscheinlichkeit von 2/3, während das Tor welches man zu Beginn ausgewählt hat noch immer die Gewinnwahrscheinlichkeit von 1/3 hat.

Diese Zahlen bestätigen sich auch durch das Programm. Irgendwie verblüffend, wie sehr die Mathematik manchmal der Intuition widerspricht.

This is our favorite news story of the year so far, since it brings together the Monty Hall Problem, M&Ms, and monkeys:

“The economist, M. Keith Chen, has challenged research into cognitive dissonance, including the 1956 experiment that first identified a remarkable ability of people to rationalize their choices. Dr. Chen says that choice rationalization could still turn out to be a real phenomenon, but he maintains that there’s a fatal flaw in the classic 1956 experiment and hundreds of similar ones. He says researchers have fallen for a version of what mathematicians call the Monty Hall Problem, in honor of the host of the old television show, Let’s Make a Deal…

If the monkey chose, say, red over blue, it was next given a choice between blue and green. Nearly two-thirds of the time it rejected blue in favor of green, which seemed to jibe with the theory of choice rationalization: Once we reject something, we tell ourselves we never liked it anyway (and thereby spare ourselves the painfully dissonant thought that we made the wrong choice).

But Dr. Chen says that the monkey’s distaste for blue can be completely explained with statistics alone. He says the psychologists wrongly assumed that the monkey began by valuing all three colors equally.”

Link (via New York Times)

Alternate Link (in case New York Times website is not dependable, as usual)

[I do not give permission for others to use this essay in any way, shape, or form without informing me. If you would like to use parts, or refer back to parts, ask me and I will consider.]

We’ve all gone through the stressful rites of learning the scientific method in elementary and middle school, these rites being accompanied by multitudes of science projects throughout the years. Eventually, it’s drilled into our heads against our will and we grow up thinking we know everything there is to know about the scientific method. But when one stops to think, one realizes that one really does not know about it, at least in one way: that is, its application to the computer science area.

Referring back to these past science fair projects, one will find that we were only required to build volcanoes, to plant flower seeds upside down and see which direction the plant grows, to analyze the amount of pollution in a certain area (though that was a far reach for a 6th grader), or to see how much bacteria grows in the kitchen sink. None of these have to do with Computer Science, only with the Natural Sciences. So where do the Computer Sciences really come into play? And how does the scientific method help when it comes to computer science?

The scientific method is a basic process which we use in order to come to a scientific conclusion about something. Essentially, it is used to solve a problem of some sort. It follows basic steps, although they may not be carried out exactly as I am about to list them. Firstly, there must be a form of observation of the subject. This may involve doing simple research, or pinpointing exactly where the question lies. The question will then be transferred over to the hypothesis, which is an educated guess of what the outcome of the scientific process may be. After identifying the hypothesis, one must decide how best to go about answering the hypothesis, and must formulate the outline of an experiment. Afterwards, one must actually perform the experiment as planned in the previous step. The results of this experiment must be gathered, organized, and analyzed, at which point one is ready to form a conclusion and report one’s results (Shrake, Elfner, Hummon, Janson, Free, 2006, 131).

The scientific method is relevant to our lives because it allows any person, even young children, to follow a set method that reveals how to go about solving a problem. Although one person may initially conduct a certain experiment, if that person follows the scientific method and leaves detailed descriptions of the process, then anyone would be able to go back and recreate that experiment. In this way, science can be shared with the world and not just a few people (Shrake, et al).

So how is the scientific method related to computer science? Firstly, computer science is derived from mathematics, and is also related to logic as well. In fact, computer science could not exist were it not for mathematics. It is, at its base, a study of algorithms (Knuth, 1974, 325). It is inarguable that mathematics do not include usage of the scientific method, because the scientific method is a step by step formula practically, on how to solve a problem. Mathematics works the same way. In a sense, every time one does a math problem, one is following the scientific process to a “t.”

Computer science, although most closely related to mathematics, can also be related to most other sciences, in that one can figure out chemistry, music, physics, and many other sciences through the use of computers or computer science in general. Chemistry is a natural science, and thus implements the scientific method throughout its area. So by proxy, the computer sciences do use the scientific method, for a step-by-step implementation of whichever scientific area they are dealing with.

One can get even more specific, however, with an explanation of how the scientific method is directly used with computer science. The scientific method requires specification and even computer science can fit the bill. For example, a popular computer science experiment among students is the Monty Hall Problem (Braught, Reed, 2002). This is in reference to the host of the tv show “Let’s Make A Deal” in which there is a prize hidden behind one out of three doors and the other two doors have nothing behind them. A contestant can pick one door, but is then given the opportunity to switch doors. The problem lies in whether it is in fact more beneficial for a contestant to pick a different door or stay with his initial pick.

Following the scientific method, one would first determine the problem, described above, and observing it, which would involve looking at how the game is set up, doing basic analysis of the probability needed, etc. A hypothesis must then be formed, using what one theorized during the observation of the problem to come up with an educated guess of the answer. From then, a program, a simulation of the problem, must be created that can resolve the answer. This step involves two parts: Firstly, the planning out of the program (the experiment) and secondly, the building of the program, namely, writing it and then executing it. One can then form and write a conclusion based on the results of the experiment, the running of the program (Bruaght, 108). This experiment utilizes the scientific method, as well as showing computer science at its base, thus proving that the scientific method can in fact be used in the computer science area.

Science, a complex art, will always provide an infinite gold mine of knowledge to anyone willing to dig it up, whether twelve years old or eighty. Although commonly considered a “newer” form of science, it’s existed for centuries, if only in theory rather than practice. Every science has a similarity, a bond, and that is the beauty of the scientific method, the process that lets us take a complicated idea and transform it into one that anyone can comprehend.

Works Cited

Braught, G., Reed, D. (2002). Disequilibration for teaching the scientific method in computer science. ACM SIGC SE Bulletin, 106-110. Retrieved November 21, 2007 from

Knuth, D.E. (1974). Computer Science and Its Relation to Mathematics. The American Mathematical Monthly, 323-343. Retrieved November 20, 2007 from

Shrake, D. L., Elfner, L. E., Hummon, W., Janson, R. W., & Free, M. (2006). What is science? Ohio Journal of Science, 106(4), 130-135. Retrieved November 18, 2007 from Academic Search Complete Database.

Let this stand as a testament to how the popularly and improperly taught truth about the Monty Hall Problem is false, for a variety of reasons.

If you’re unfamiliar with it, check out either this video (5min.com) or that video (YouTube). They’re two seperate explanations.

To set an official designation of the rules:
You are a contestant on a gameshow. The host gives you the option of selecting one door from an available three. Behind one door is a new car, behind another is a goat, and behind the remaining door is also a goat. You select one at random. The host now reveals one of the doors you have not selected to be a Goat Door. What is the probability of selecting the Car Door?

The disturbingly frequent misinformed explanation is that switching gives you the greater advantage.

Here is how exactly that line of reasoning is mortally flawed.

The greatest of logical errors is the perception of switching, when in fact you are not switching anything. The first choice is not a genuine choice because the revelation that one of the three doors you are permitted to pick from will be removed. You are in effect simply playing the game. Once you’ve made your misperceived selection, you are given the choice to choose between two options. This is your final decision. The original selection when there were three unknowns does not influence the final decision, because you would have had to choose between two new options regardless of whether you had picked any of the three.

The original choice was one out of three (choose the first door, choose the second door, choose the third door), but the second choice is one out of two (choose one door, choose the other door). The very question itself posed, “do you want to swap (option 1), or do you want to stay (option 2)” in itself defines the new set — the new probability, of 2 choices. Whether or not there had been 2.38 fillion dillion doors is irrespective of whether or not the new question is “stay or swap” at present.

The option between “switch or not switch” are the new options, and now has nothing to do with doors or original previous options.

The oft-cited, poorly-reasoned proof (which Wikipedia does here, is that:

The fact is, is that when you are asked to select a door in the beginning, you have made a selection that only applies to the original terms. The flawed reasoning operates under the circumstance of the original question imposed upon the second question — but the new question is what sets the standard for calculation, not the original. The reasoning fails to include the fact that the announcer will not open the Car Door AND the announcer will not open the door you have selected, which creates an actual probability of 50/50, because the 3rd door he reveals was never an actual option. Any version of the Monty Hall Problem that includes BOTH of these stipulations AND declares there to be a 66% chance of win by switching bears goat-like intelligence.

Rephrased, the most glaring problem is that there is actually only one chance, 100%, that the Car door is the Car Door. There is a 0% chance that Goat Door A is a Car Door, and a 0% chance that Goat Door B is a Car Door. Of the available options, three doors, you have a chance of 100 divided by 3 options. When one option is eliminated, there are now only two options. That makes the probability of 100% that the Car door is the Car door, and 0% that the Goat Door is the Car door. Now that you only have two options, the probability is 50%.

Even if there were 100 doors, as if often used to rationalize that switching makes better sense, there would STILL only be a 100% probability that the Car Door is the Car Door, and 0% for each of the 99 other Goat Doors. By the purported proof offered by Wikipedia, switching would therefore offer you a 99% chance of getting the door correct by switching, when in fact you are actually only choosing between one door or one other. Regardless of how many past choices you had before, the current question between “switch or swap” (2 options) is what sets the stakes.

Take again for instance if you were asked to select 2 of the 3 doors:

1. You pick the Car Door and a Goat Door, a Goat Door is revealed to be the third. You have a choice between your original selections, a Car or a Goat — a 50/50 shot.
2. You pick a Goat Door and the Car Door, and a Goat Door is revealed as one of YOUR doors. You have a choice between your remaining unopened door or the unselected door — a 50/50 shot.
3. You pick a Goat Door and the other Goat Door. The announcer will unfailingly reveal one of YOUR doors as a goat door. You still have a choice between your remaining unopened door or the unselected door — a 50/50 shot.Take again for instance if you were to select one door and the host were to reveal that YOUR door is the Goat Door. Your probability is still a 50/50 chance of your new choice correct. The fact here is that you must make a new choice, which is identically probable to the dilemma of whether the host had NOT opened the door you first selected — because the choices were between STAY or SWAP.

Take yet again for instance if you were to select one door and the host were to reveal that one of the doors you had not selected to be a Goat Door. Now, you are turned around and blindfolded, and the prizes behind the doors are rotated behind the two available doors, so that you’re unsure whether the door you had selected is actually still corresponds to your original Car Door guess. The probability does not change, because there are still only two options. The chances are still 100% that one door will be correct and 0% that the other door will be incorrect, resulting in a 50% chance.

Take yet again a visual illustration of your choices: To pick between three doors is like unto throwing a dart at a rotating (a plane rotation) circle that has three equal sections defined as section 1, 2, and 3. It is revealed that the corresponding door the dart hits is your first selection, and following is revealed that one of the unhit doors is a Goat Door. The new circumstance is like unto throwing the dart at a new circle divided half and half into “stay” and “swap” respectively. You are not throwing a dart at a plane-rotation circle with all three options, because you are clearly not going to select the already-opened goat door. The new selection is the identical odds as whether you had not even selected one of them originally.There is no possible way to discern whether the door you have already selected is the Car Door, and LIKEWISE is there no possible way to discern whether the unselected door is the Car Door. However, it IS possible to know whether the third door is a Car Door, because it has already been revealed as a Goat Door. The options are now only between one unknown and another unknown — 50/50.The reason the purportedly true answer gains acceptance is because it stands to reason only in one case, and not the rest. This is akin to the assertion that the scientific method is the only way to determine truth. In order to determine that very statement, you would have to establish the fact without using the scientific method. It’s the same “How do you know the bible is true? God says so! Where? In the bible!” argument.

Wikipedia reasons, This difference can be demonstrated by contrasting the original problem with a variation that appeared in vos Savant’s column in November 2006. In this version, Monty Hall forgets which door hides the car. He opens one of the doors at random and is relieved when a goat is revealed. Asked whether the contestant should switch, vos Savant correctly replied, “If the host is clueless, it makes no difference whether you stay or switch. If he knows, switch” (vos Savant, 2006).

This is also incorrect, because the actual probability of one door being 100% the Car Door and the other door having a 0% chance of being the Car Door remains true, regardless of whether the host knows or not. To Switch (option one) or not to Switch (option two) is what is being asked. You are being asked to choose between two actions, the original probability now being completely irrelevant.

Wikipedia also cites a graph showing three doors, where the prizes are actually revealed. It lists the Car Door under a 33% bracket, and brackets the two remaining doors under a collective 66% deliniation (a Venn diagram). The problem here is twofold, at a minimum. Firstly, when you divide 66% by two, you do not have 66% remaining. The two doors do not represent a 66% chance each that averages ([66+66]/2) to 66%, because each door is purportedly a 1-in-3 chance each, or 33% each. Secondly, there is not a 33% chance that the Car Door is the Car Door — it is 100% the Car Door. The other two options are 0% and 0% respectively. If you were to eliminate one of the 0% probability doors, a Goat Door, you’re left with either the 100% option or the 0% option, making it a 50% chance.

The diagram found here uses inverted teacups concealing a diamond. The diagram makes the assertion that switching your original choice when one is revealed. However, the actual choice being made is between one cup or one other cup, a 50/50 chance. The original formula does not apply since one cup has been eliminated. It’s a false positive.

Playing the game yourself and compiling the results is completely arbirtrary because at what point do you cease the experiment? If you were to select the Car Door in your original 33 selections without switching, could you simply just quit, sufficed that staying with your original guess is the 100% sure strategy? It’s the same question of asking whether a coin flipped 50 times resulting in 50 heads will next flip tails.

I hope you have begun to see reason more clearly if you had been or still are a “66-percent-chancer” in regard the Monty Hall Problem. Please feel free to discuss below =)

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