So today is the twenty-third of October. Those of you who pay any attention whatever to the field of cosmogony (or read Pharyngula) will know that today marks a very important date. Yes, according to Bishop Ussher, today is Earth's birthday—today, our planet (and, indeed, all of the cosmos) turns 6013. Also, as Terry Pratchett and Neil Gaiman pointed out, the Earth is a Libra.*
Have any of you ever heard of the Monty Hall problem? It's based on the television show Let's Make a Deal, and it goes something like this:
You're on Let's Make a Deal, and Monty Hall has offered you a chance to win big: you're asked to select one of three doors. Behind one of the doors is a fabulous new car! But behind the other two are hilarious gag prizes that no one would ever want!
You point your finger toward one of the doors, trembling with excitement. Monty Hall smiles, steps up to one of the other doors, and opens it, revealing a goat! And then, with a twinkle in his eye, Monty Hall offers you a choice: you can either keep what's behind the door that you originally selected or you can switch to the third (unrevealed) door.
Now, here's the question: What should you do?
I've heard that even mathematicians argue about this one, because the solution is so contrary to common sense. [If you're really dying to know what the solution is, click here.]
But that's the problem: common sense only gets you so far. You've probably heard it said before that "Common sense is neither common nor sensible." It's great for the little things, like determining whether your brother split the piece of pie fairly or roughly how long it will take to get to grandma's house—but when common sense fails, it can fail spectacularly. Try using common sense to ascertain the shape of the earth, the motions of the planets, the age of the universe, the origin of life, the causes of and cures for disease, or even the solution to a fairly simple math problem, and the whole thing falls apart: you either get an answer that is spectacularly wrong or spectacularly useless. In fact, I would wager that the entire scientific enterprise is built upon moving beyond common sense.
I really enjoy figuring things out. I love learning something new. Our species thrives on innovation—that really seems to be our niche.
Have you ever heard the old proverb "curiosity killed the cat"? I hate that saying. When I was a kid, I asked a lot of questions. I imagine that I wasn't alone in this. Anyone have kids? They're curious, right? Right. So I asked a lot of questions. I was mostly raised by my father, and he was—is—a great dad. (Sure, he had some strange ideas—but moms and dads and aunts and uncles and brothers and sisters and husbands and wives and children and friends and pretty much everyone who isn't you is going to have some pretty strange ideas, right? You deal with it.) But my dad is a really great dad, and one of the many really great qualities that really made that really great dad great was the fact that he always, without fail, every time encouraged my curiosity. And that is, if you'll excuse me saying so, really great. If I asked questions, he'd answer them, but he'd also ask questions back. He made me think about things, and no question was ever off-limits.
I think that curiosity is really important. But... not everyone agrees with me. Try this quotation on for size:
"There is another form of temptation, even more fraught with danger. This is the disease of curiosity. It is this which drives us to try and discover the secrets of nature, those secrets which are beyond our understanding, which can avail us nothing and which man should not wish to learn."
Anyone know where that quotation comes from? That's from Saint Augustine's Confessions, and I like it in probably exactly the same way that Professor X likes Magneto. That kind of thinking is very dangerous. It leaves people dying of plagues and famines, it stifles innovation, it encourages an insular society closed to new ideas, it discourages free inquiry into the mysteries of nature, and ultimately it can avail us nothing.
This isn't about religion, so I'll put this train back on its tracks in just a second, but I want to share one more thought with you on the subject. Julian Begini expressed this quite well, I think: "It is arguable," he said, "that humanism has a better grip on life's mysteries than religion. For example, I'm genuinely in the dark about how the universe started, whereas plenty of religious believers have that hole in their understanding plugged by their deity."
I'd rather not know than have a non-answer. To me, saying "God did it" is like saying "it's magic!" You're not actually answering the question.
Okay, that's enough about that: I don't want to get preachy. Nobody likes preachy, right?
You know, I briefly considered calling this talk "The Disease of Curiosity". I frankly thought that would be a brilliant idea. Thankfully, my wife talked me out of it. We now have fewer talks that sound like they're about contracting a venereal disease.
Okay. Let's learn something!
You know what I learned? I learned that despite being a happily married heterosexual man, I have a huge crush on Adam Savage.
I'm the lead developer of a Winnipeg software company specialising in machine learning applications. As part of my job, I have to interview many prospective full-time, part-time, and co-op employees. I conduct about twenty interviews each year. These people are smart people, and the positions that they're interviewing for are challenging ones—many are going to be working in one or many programming languages with which they have little to no experience, they'll be working on cutting-edge expert systems, and, quite frankly, they won't be paid very well. For this reason, the interview process that we employ is what can only be called grueling.
These people's job will be to figure stuff out. You probably won't be surprised to learn that I prize the ability to solve a novel problem more highly than I do a knowledgebase. If an applicant tells me that she has a working knowledge of Scilab, Python, FreeMat, PERL, and SQL, that's great! But it won't get her a free pass. And so, as a warm-up, I like to ask a few riddles.
You are presented with four cards lying on a table. Each card has a number on one side and a letter on the other, and the visible faces read "A", "B", "2", and "3". You are provided with an hypothesis, and it is this: each of these cards that has a vowel on one side has an even number on the other. You have permission to flip over two of the cards. If you want to conclusively confirm or disconfirm this hypothesis, which two should you flip?
I'll let you think about that for a moment. Once again, the hypothesis is: each card with a vowel on one side has an even number on the other.
Poll the audience for each of the six solutions: A & B, 2 & 3, A & 2, B & 3, A & 3, B & 2.
This problem is pulled directly from the pages of John Allen Paulos' Innumeracy, a wonderful and eminently readable book from which I try to plagiarise at least once a day.
One of the things that this problem does very well is showcase confirmation bias: that is to say our tendency to look for confirmatory evidence and ignore disconfirmatory evidence. While many will opt for flipping "A" and "2", flipping "2" will actually add no new information to the system. Remember that the hypothesis was that each card with a vowel on one side has an even number on the other, NOT that each card with an even number on one side has a vowel on the other. If "2" has a vowel on the other side, it is consistent with our hypothesis but does not confirm it, as "3" could still have a vowel on its reverse; if, however, "2" has a consonant on the other side, it will neither confirm nor disconfirm our hypothesis because it does not fit into the problem space. "A" and "3" are the correct cards to flip, because in all cases revealing their other sides will positively confirm or disconfirm our hypothesis.
Anyone remember the Infinite Improbability Drive from Hitchhiker's Guide? The basic idea was that if you could calculate precisely how improbable it was that you would spontaneously appear somewhere else, you could do just that. It also had the nasty side-effect of causing incredibly improbable things to happen.
Shall we make something really improbable happen? Let's talk about numbers for a bit. I work with numbers all day, and I love 'em. They're weird little monsters, though. I asked my computer to generate two sequences of twenty-five (pseudo)random numbers between 1 and 10, and then I sorted them so that you could easily see the distributions.
3 7 2 2 9 3 7 4 7 5 9 3 10 2 6 9 3 3 6 8 3 2 6 10 10
3 2 6 5 6 8 7 4 5 3 10 8 4 7 8 3 5 9 1 1 1 3 10 8 2
2 2 2 2
3 3 3 3 3 3
6 6 6
7 7 7
9 9 9
10 10 10
1 1 1
3 3 3 3
5 5 5
8 8 8 8
Which of those number sequences looks more random?
What do you notice when you look at the numbers on the left? The first thing jumped out at me was that there were no ones! The probability of selecting 25 random integers between one and ten and not receiving any ones is only about 7%! Did I make a mistake when I input the query?
Did you catch what I did just there? It isn't actually any more likely that I would arrive at that particular sequence of numbers than that I would pick seven every single time. Prior to the numbers being picked, the probability of arriving at any number sequence with that distribution of ones and twos and threes and so-ons was about one in 200 million, and the probability of arriving at that specific sequence of numbers was one in 10 septillion.
Afterward? The probability was one in one. It's important to remember that the chances of arriving at any random sequence of numbers is the same, and anything that was picked would be equally improbable. So next time someone tells you about some "one in a million thing" that just happened to them, you may have cause to be less impressed. Astoundingly improbable things happen all the time.
Now, I have to admit something: I lied to you all a moment ago when I said that I asked my computer to generate two sets of numbers. To my eyes, the sequence on the left doesn't look nearly as random as the sequence on the right. There are way too many threes, there are no ones, and three of the numbers were only selected a single time! Those are the things that I tried to correct when I hand-crafted the list on the right. Contrary to our expectations, randomness can actually be remarkably clumpy.
For those of you who are interested, I calculated the probability of arriving at that numeric distribution binomially, and it's entirely possible that I made a mistake. Feel free to work through it yourself. You'll need these equations, which you may remember from high-school pre-cal:
Pr = nCr · pr · (1 - p)n – r
nCr = n! ÷ (r! · (n – r)!)
When you watch MythBusters, what is it that draws you in?
For me, it isn't the explosions. I love to watch Adam and Jamie trying to figure things out. Why? Well, partly it's because I want to know the answer, and I want to see how they intend on arriving at it. But more than that, I love to watch how excited they are. It's contagious! Be excited about learning. I'm sure that most of you can remember a teacher that you had in high-school who was really enthusiastic about his or her subject matter. That's a wonderful thing!
And as media expands, those people can rise to the top. Today we have Adam Savage, Jamie Hyneman, Phil Plait, Neil deGrasse Tyson, Brian Dunning, Rebecca Watson, Simon Singh, Richard Wiseman, the Novella brothers... the list seems endless.
Carl Sagan isn't around, anymore. Neither is Richard Feynman. But their legacy lives on. We need people to be excited about learning and about problem solving and about science, because we don't want our society to stagnate.
How about one for the road? This is a problem that is fairly well designed to confound computer scientists, and it's currently my favourite puzzle. Let's say that you are given a balance scale and eight weights. The eight weights are identical in appearance, but one is very slightly heavier than the rest. You can use the balance scale twice, putting any number of weights on each side and observing the result. Is it possible to conclusively determine which of the eight weights is heaviest in this way?
I'm not going to reveal the answer to you. If you think that you've got it, feel free to seek me out later, but please don't announce it to everyone, because you'll spoil the fun.
So what am I trying to say? What it really comes down to is this: this disease of curiosity against which Augustine railed has led to all of the amazing advancements in science and technology of which we avail ourselves on a daily basis. Curiosity is pretty much the best thing we've got going for us.
When you spend a lot of time trying to figure things out, sometimes you just won't get it. And that's okay. There have been times that I've literally spent five hours trying to solve a particular mathematics problem, for no other reason than that I wanted to know the answer. Sometimes, you're just not going to figure it out—but that's no reason not to try. I'd always rather see somebody fail than see that person not try at all.
It's a cliché, I know, but from failure you learn a whole lot more than you do from success. In Last Chance to See, Douglas Adams said that "human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so." Let's prove him wrong.
Oh. And you should switch.
* Technically, in 4004 BCE, I believe that the Earth would be a Scorpio, although 23 October is firmly Libra in today's sidereal astrology. But for some reason, Libra makes the joke better. This explanation, however, does not.
Spoiler alert! If you don't want to learn the solution to the Monty Hall Problem, read no further!
Do you think that it doesn't matter either way? Each door has a 50% chance of having a goat on the other side? Well, if so, then you'd be wrong. You should switch doors.
It's important to remember that Monty can and will always reveal a goat. Since we're dealing with a relatively small problem space, let's explore it. First, we'll assume that you decide you'll always stay. We'll label the winning door D1. There are three possibilities:
- You pick D1. He reveals a goat (either D2 or D3). You stay. You win!
- You pick D2. He reveals D3. You stay. You lose!
- You pick D3. He reveals D2. You stay. You lose!
So if you stay, one time in three you'll win.
Next, assume that you decide you'll always switch. Again, we'll label the winning door D1. There are three possibilities:
- You pick D1. He reveals a goat (either D2 or D3). You switch. You lose!
- You pick D2. He reveals D3. You switch to D1. You win!
- You pick D3. He reveals D2. You switch to D1. You win!
If you switch, two times in three you'll win.
A less intuitive (more "mathy") way of explaining it is that the door that you pick initially has a probability of ⅓ of being a winner. The other two doors, taken together, have a collective probability of ⅔ of having a winner among them. When Monty reveals that one of them is a goat (which he always will), that set of two doors still has a ⅔ chance of containing a winner. Since you know that one of them is a goat, the other has a ⅔ probability of winning.