From a draft started on 11 June 2008:
I’ve always had a keen preference for most things mathematical. In school, my favourite subjects were always Math, Algebra, Physics, Chemistry, Calculus, and Geo-Trig. I had a high proficiency for it and, not surprisingly, that was why I liked them so much. I kept being shocked and thne thrilled at how much overlap there was and how many interdependencies there were between the different studies, and how it all moved like a well oiled machine.
That’s not to say I found all Mathematics enjoyable. Perhaps surprisingly, given my current field of work, Logic was a subject I struggled with. Oh, I was fine with the symbols and equations, but as soon as the curriculum started mixing in the ambiguities of the English language, my comphrehension level lowered quite drastically. I have no problems with spelling or grammar (I just get lazy sometimes), and will even correct people almost to the point of them threatening physical violence. 
I have no problems rattling off coherent thoughts in essay form, but as soon as you try to adhere it to a logical argument, for some reason, my brain turns to cabbage. Oh, I can fake it very well, but I generally only do so for comical effect.
The higher levels of Linear Algebra also give me palpitations. I struggled all through my last Linear Algebra class in university. I understood the basics. I grasped the concept of matrices in their basic form and the equations they represented. But at a certain point, which I still haven’t been able to locate, a disconnect grew between the matrix manipulations we were performing and the underlying equations we were trying to solve. Jiggery-pokery with pivot columns turned into hocus-pocus and I suffered through the rest of the class just hoping to make good educated guesses.
The last subject I had trouble with was Statistics and Probability. I saw, and in most respects still do, Statistics as no less a pseudo-science or voodoo than Astrology, Alchemy, or Phrenology. I’m guessing a career in Quantum Mechanics is not waiting for me. I’d include Intelligent Design in that list of outlaw sciences, but it’s not even a pseudo-science, it’s a religion (but that’s for another time).
Statistics seems to be one of those sciences that is specifically designed to be counter-intuitive, and any similarities to common sense is purely by chance (no pun intended).
Which brings me to my point. I just saw the movie 21 last night and, needless to say given its subject, it involves some math-related exposition. One such example near the beginning of the movie is when Kevin Spacey’s character, Professor Micky Rosa, proposes a problem he calls the “game show problem” (more commonly known as the Monty Hall Problem or possibly also the Three Prisoners Problem), which basically goes as follows:
A contestant is presented with three doors and is asked to choose one. One of the doors hides a reward, while the other two do not. The host knows what lies behind each door. Upon making his/her choice, the contestant is shown what’s hidden behind one of the doors not chosen (the prize door will never be revealed). The host then gives the contestant the option to switch his/her choice to the only remaining unrevealed door. The question is: Is it in the contestant’s best interest to switch the choice?
Spoiler: Common sense tells us no, it doesn’t matter. It’s a 50/50 shot.
Mathematicians, however, would tell you that it does matter. In fact, they would tell you that you should always switch your choice. They would tell you you have a 66% chance to win the prize.
Huh?
This is also the conclusion reached in the movie as well. Being a fairly well-known problem, and one that flies in the face of common-sense, countless explanations are offered with varying attempts at explaining it simply, none of which seemed to give me any satisfaction. Though I’ll admit, the second table in the Aids to Understanding section of the Wikipedia article explain it quite well. The trick of the original problem is that the host knows where the prize lies, and at times his hand is forced depending on the original choice of the contestant.
I’m anything but an expert on probability and statistical analysis, but here’s how I arrived at a solution that makes sense to me. My first reaction, being a programmer, was to build a simple program that tooks a couple of inputs (numbers of choices, i.e. 3, and number of iterations, i.e. 10 times, 100 times, 1000 times, etc…), define an array to act as doors, use a random number generator to shuffle the position of the car and to make the contestant’s decision, reveal empty doors (as per the stated host’s behavious), and then tally the number of times the contestant wins by switching his choice.
Rather than think it all out into an algorithm ahead of time, i just took to the keyboard and started writing code, adjusting for the rules of the game as I went along. After a few minutes, it came abundantly clear to me why the mathematicians were correct. In a roundabout way, my program essentially aimed to disprove that always switching your choice is more beneficial.
Because the results of the game are either you win the prize or you don’t, I found it easier to think of the problem in terms of its complement. Going with the assumption that we are always going to switch our answer, the only times we’ll lose is when we initially chose the door with the prize. So because the chance of choosing the door with the prize is 1/3, the chance of winning is 1 – (1/3) or 2/3.
Now maybe that’s the wrong way to think about it, but like I said, probability’s never been my strong suit.
I wonder if any studies have been done regarding Deal or No Deal…
