No real reason for posting this other than an interesting essay on a mathmatical concept. It has some implications when discussion evolution, of course. Part of a course on
Joy Of Thinking: The Beauty and Power of Classical Mathematical Ideas. http://www.teach12.com/store/course.asp?ai=18879&id=1423&d=Joy+of+Thinking%3A+The+Beauty+and+Power+of+Classical+Mathematical+Ideas-Dave
Random Thoughtshttp://www.teach12.com/ttc/InternalFigs/1423_22.asp?ai=18879by Professor Michael Starbird
There's a wonderful, classic example of randomness that I am sure you have heard of, and this is a question about monkeys typing at the typewriter. The question is: Suppose you had monkeys typing randomly at the typewriter, completely randomly, hitting the keys at random, not knowing what they mean. If we have enough of them, and let them go long enough, would they eventually produce Hamlet?
This very famous question, in fact, is one that goes back to Sir Arthur Eddington, the famous astronomer. He made this observation back in 1929, and he was trying to describe some features of the second law of thermodynamics, and he wrote the following: “If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys was strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly favorable.”
Well, in fact, he has the correct answer. In particular, if we have enough monkeys typing randomly at keyboards, what we will see is a perfect version of Hamlet. Now, that seems so incredible. How could randomness actually lead to such a famous book?
We can illustrate this and make it more intuitive by considering a simpler issue. Let's actually boil down Hamlet to just one number. Let's suppose that Hamlet was just one number. Instead of lots and lots of words, and lots of footnotes, and lots of characters, and so forth, and scenes, let's just suppose that Hamlet was the number 3, and suppose that I had a die. This is now my keyboard, and I have a die, and I start rolling it. The question is, if I do this long enough, will I see a 3? Well, the answer is “certainly,” assuming this die is a fair die, so that all sides are as equally likely to come up as any other. We know that, in fact, the answer is “yes.”
Now, how do I know that? I'm going to prove it to you in a couple of different ways. First of all, just think of every time you have either rolled a die, or watched someone else roll a die. At some point in your life, you saw a 3, so that proves it, right? You all saw a 3. If Hamlet, then, were just the number 3, then it is very intuitive that if we kept rolling enough, or let the monkeys randomly hit the 6 keys, we would expect at some point that some monkey would hit 3, or, in our case, to roll a 3, and then, we would have Hamlet.
Great. From a mathematical point of view, once we have this idea, the whole issue is resolved, because all we have to do now is to make it a little bit more complicated, so instead of looking at Hamlet being just one number, let's suppose that, in fact, Hamlet is two numbers. Let's suppose that [I have two dice, one yellow, one green, and] Hamlet is “yellow 1,” and “green 2.” That turns out to be Hamlet. That is a little more complicated. In fact, how many possibilities are there if I were to roll these two dice? There are six possibilities for [the green die], and six possibilities for [the yellow die], so if you look at all possibilities, you see that there are 6 times 6, or 36 total equally likely possibilities. The likelihood, then, of getting a yellow 1 and then a green 2 turns out to be 1 out of 36, so far less likely than just rolling a 3 on one die.
However, 1 out of 36, if I were to wait long enough, and keep doing this, what would you guess? Well, you would guess that if you keep rolling these dice enough times—oh, look at that, 3-3; that's not Hamlet—you would expect that maybe that's Julius Caesar or something. I don't know, but that's not Hamlet—we have to keep rolling. If the probability is 1 out of 36, though, which is slim but still possible, if we repeat that enough times, we would expect to see a 1 and a 2. Do you think, in your own life experience, watching people roll dice, you will remember that you once did see a 1 and a 2? That person produced Hamlet, if Hamlet were only two numbers.
To the mathematician, that answers the question, because now, what do we have here? Now, we just have a die, if you will, with lots and lots of numbers on it; in particular, one side of the die for every letter, every character, and a space, and a period, and so forth. So imagine that really big die with, maybe, 48 symbols on it, since there are 48 symbols on a standard keyboard. Imagine a 48-sided die, if you will, and then you start rolling it and writing down what you see. Every letter corresponds to what we write down.
Really, then, Hamlet is just a really long, long, long, long sequence of dice rolls, and it is extremely unlikely that you're going to roll it billions and billions of times and produce the precise sequence of rolls that spells out every single letter, every single space, in Hamlet, but that possibility, in fact, exists. It's not 0. It's very slim, and if you have a very slim experiment, in terms of the chances that it occurs, the chances are very slim that if you repeat that long enough, you will expect to see it. Just ask. If you repeat this long enough, you will see a 3. It is the exact same idea.
In fact, then, if you let the monkeys type away, what we do see at some point is Hamlet. The thing is that sometimes you might actually have to wait quite a while to see Hamlet, right? How long would you have to wait? Well, it turns out that you can actually use some calculations. Suppose you took just a mere million monkeys. Of course, in my mind, when I do the experiments, I have infinitely many monkeys, and they are typing forever. Let's just take a million monkeys, and let's suppose that the monkeys are hitting the keys at a rate of one keystroke per second, so that's pretty fast.
By the way, when we do this experiment, in our minds and in Sir Arthur Eddington's mind, we genuinely mean that the keys are being hit at random. If real monkeys try this… In fact, they tried this in England, believe it or not, some time back. They actually did an experiment, and it turns out that the monkeys might have been enamored with the “S.” They hit the “S” a lot. They liked that key. That's not random. When we mean monkeys typing at the typewriter, we mean that every key is just as likely to occur as any other key is.
Suppose they are typing at one key per second, and we have a million of them going. Well, there are a lot of possibilities. How long might we have to wait in order to see someone type out, “To be or not to be. That is the question.”? How long would we have to wait? Probably more than 1060 years. That's 1, followed by 60 zeros. That's how many years we would have to wait in order to expect to see just the phrase, “To be or not to be. That is the question.” That seems like it's pretty slim, and it is, but if you wait that long, we would expect to see that phrase. If we wait even longer, we would expect to see more, and if we wait forever, we would expect to see, in fact, all of Hamlet.
It reminds me, actually, of a wonderful old routine credited to Steve Allen, the very first host of the Tonight Show, long before Jack Paar or Johnny Carson and Jay Leno. The routine was basically something like this: You see Steve Allen, and he's pretending to be a reporter. He says, “Okay, well, scientists have said that if we have enough monkeys type long enough, they will finally bash out Hamlet. Well, actually, scientists are trying that experiment right now, and let's see how they're making out.” Then, there's somebody dressed up in a monkey suit hitting the typewriter, he pulls out the piece of paper, and reads, “To be, or not to be. That is the zymfanblot. So close,” and then he tosses it back, and continues, “Back to the studio.”
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