A few years ago, I was reading a book called “The Curious Incident Of The Dog In the Night-time”. It’s a fun little book. In that book, the lead character talks about a problem discussed in a real world incident. That problem, sometimes called the Monty Hall Problem, is this:
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?
My immediate response was that it shouldn’t make any difference. However, the person, Marilyn vos Savant, who answered the question, said that contestant should switch to the other door. The reason she stated was
The first door has a 1/3 chance of winning, but the second door has a 2/3 chance.
Now, this seems counter-intuitive at best, and absurd at the worst. And, it seems that I was not the only one who thought so. She got thousands of mails, some of them from the PhDs in mathematics, claiming, sometimes in not-so-polite words, that she was wrong. Here’s a sample of a letter from a condescending PhD:
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble.
Everett Harman, Ph.D. - U.S. Army Research Institute
And another one from an irate, self-righteous mathematician, also a PhD:
You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!
Scott Smith, Ph.D. - University of Florida
Oh, that was my reasoning too. And yes, Marilyn was once listed in the Guinness book as the person with the highest IQ in the world. You can go to Marilyn’s own page documenting the whole controversy. It might seem like an overstatement to call this a controversy, but if you get like 1000 mails from mathematics PhDs only, I think it counts as one, at least in the field of math.
However, it turned out that Marilyn was indeed right. She got a mail from a PhD at MIT which went something like this:
You are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong!
Seth Kalson, Ph.D. - MIT
Let me try to explain why it is right with some help from Wikipedia.
One way this can be explained is that if the contestant picks a goat initially (66.66% or 2/3 probability), he/she will win the car by switching since the second door with a goat is now eliminated by the host. The only remaining door is the one with the car. On the other hand, if the contestant did choose the car-door initially (33.33% or 1/3), switching the door will not help. So, If you always switch, the probability of the first scenario, 2/3, is also the probability of you winning the car.
I don’t want to get into more details here. The Wikipedia entry contains several different kinds of explanations, including graphical and mathematical ones.
Mathematics has a lot of such unintuitive, quirky stuff that can stump you. Here is one more example.
0.9999… and 1 are the same numbers
The … after the 0.9999 mean that the 9’s continue forever.
I am not saying that they are equal upon rounding or that they are practically the same. I am saying that they are different representations of the same number, just like 1/2 and 0.5 are different representations of the same number.
Really understanding, and accepting, this fact requires coming to terms with the concepts of infinity and infinitesimals, and some understanding of the real number system. However, the basic proofs are simple enough to understand. Here’s one that can be understood by school kids:
x = 0.999... 10x = 9.999... // multiple both sides by 10 10x - x = 9.999... - 0.999... // subtract x from both sides, remember x = 0.999... 9x = 9 x = 1
See, exactly the same. As usual, the Wikipedia entry has a list of several proofs, even more “rigorous” ones, if you are interested. And, if you want a really fun explanation, here’s a video from my favorite recreational mathemusician.
Most people find this hard to believe; even with all those proofs. That could be because imagining infinity is really hard. However, even more basic reason, I think, is because most of us weren’t taught math like it is supposed to. The hardest thing I did in trying to learn math was getting out of the school mode. We assume that any number has only one representation as a decimal. Well, as the The Straight Dope says:
The lower primate in us still resists, saying: .999~ doesn’t really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.
You may encounter more bizarre results and conclusions as you go exploring more and more math. However, along with some mind bending stuff, math delivers some fantastically beautiful results.
Euler’s Identity, aka the most beautiful equation in mathematics
I think most people will recognize the symbols in that equation, even if they don’t really understand them, or, as in the case of `i`, find them unpalatable.
Even then, let me attempt to summarize why it seems so jaw-droppingly beautiful. It takes three seemingly unrelated symbols
- `e`: The Euler number, the unit rate of growth, and the base of natural log
- `π`: The irrational and transcendental ratio of the circumference of a circle to its diameter, and
- `i`: The imaginary number, the square root of -1.
and combines them with three fundamental mathematical operations of addition, multiplication and exponentiation, and lives to tell the tale.
This is like some mad scientist tried to combine the genes of the Joker, Sauron and the Galactus to create the ultimate super-villain and instead, what came out was a super-hero with high cheekbones who restored the peace in the universe.
This equation has been a source of a lot of research and discussions. Mathematicians have written entire books dedicated to it, including one that says: Dr. Euler’s Fabulous Formula - Cures Many Mathematical Ills.
To confess the obvious, I do not understand the importance of this equation yet. However, that doesn’t mean that I cannot admire how unlikely and unintuitive it can seem and still be correct.
And math seems to be so full of things that look paradoxical, unintuitive and downright incorrect; so much so that even real mathematicians are stumped sometimes. But, that is also what makes math so much fun, like this limerick says:
I used to think math was no fun
‘Cause I couldn’t see how it was done
Now Euler’s my hero
For I now see why zero
Back to learning some more math.