"This one may be more philosophy than math, but... Is it true that any occurrence, however unlikely, does have a percent chance of occurrence above zero?" This is a very good question and is closely related to how mathematics reflects the "real" world. In particular, what is the meaning of "probability" or "chance" when there are infinitely many possibilities. Basically, the real world as we currently understand it does not comprise an infinite number of "things", though it is plenty complicated enough. It is not at all clear that the universe is actually infinite in extent or time: we simply don't know and may never know.
Quantum mechanics (QM) is one of the most successful scientific theories ever propounded: it has passed every test that scientists have thrown at it. Yet central to QM is the idea that energy does not flow smoothly, but is broken up into discrete "packets". Recent experiments and theory in QM also suggest that space and time itself are similarly discrete and not infinitely smoothly flowing.
Yet, traditionally, the mathematics that we use to study the world is based on calculus and other theories which use "smooth" concepts like continuity and differentiability. Consequently, mathematical techniques and their "result" must be applied and interpreted carefully. We'll come back to physics later, but let's look at a simple example from the world of mathematics and numbers, which is supposedly less complicated.
Suppose you are asked to pick a number from 1 to 10. If you are restricted to whole numbers, then there are only the ten choices 1, 2, ..., 10. The chances of my guessing your number is 1 in 10 or 1/10. Suppose now that you are allowed to choose either whole numbers or fractions. Then you might choose, say 3/19 or 1357/589 etc. What is the likelihood that I guess your number? Technically it is 0! There are infinitely many different numbers you might choose, and I only have 1 guess. Yet, it is clearly possible that you chose 1357/589, and that I guessed it. What's going on here?
This is a good example of how mathematics must be applied carefully in order to make sense. What does it mean to "choose" a number here? How much time do you have to make your choice? If you have only, say, 5 minutes, how many digits could you think up or write down or type, or otherwise specify in 5 minutes? There is certainly some limit in size for the numerator and denominator, so there are really only a finite number of numbers that you could realistically choose in the 5 minute deadline. Even if you were allowed a day, or a week, or a year, or your whole lifetime, there are only a finite number of numbers you could specify with total accuracy simply by giving the digits of their numerators and denominators. But wait: there are other ways of specifying whole numbers. We could use "exponential notation". For example, there is the whole number 106847-1: a number consisting entirely of "9"s: namely 6847 of them. That's a pretty big number written quite economically. Let's call it N. Now we can take N and raise it to the Nth power: NN. Wow: that's pretty big yet easily written. Clearly we can continue to raise increasingly larger numbers to increasingly larger powers. Furthermore, we might come up with even more clever ways of denoting larger and larger numbers more and more efficiently. Now how large a number could one come up with in 10 minutes, or a week, or a lifetime?
Mathematicians have, in fact, played this game. My friend GS tells me that, in a coffee house a number of years ago, he and some other friends, including a few very prominent mathematicians, tried a competition: each would write, on a sheet of paper, the largest numbers he/she could express. After a time limit of 10 minutes or so, they would compare papers and try to estimate the largest number expressed, which would be The Winner, and the smallest would be The Loser. One mathematician, in order to avoid being The Loser, simply wrote: "GS's number, whatever it is, plus 1." Some other mathematicians also tried cutesy tricks. The winner, GS claims, was an artist who had simply spent the 10 minutes writing, in minute script, hundreds and hundreds of digits...
believes in the ingenuity of the human brain, so he/she thinks that there is
no limit to the size of the denominators or numerators, so there are infinitely many numbers that one could think of. Does this make the probability of my guessing your number 0?
Not necessarily! The reason is that even though there are infinitely many possible numbers,
they don't all have equally likely chances of being selected by you. Which number is more likely to be the one you think of, 3/16 or 10
6847 - 1 divided by 16889
22601+17? The more complicated the number, the more unlikely it would be that you thought of it. Really complicated and bizarre fractions have vanishingly small probabilities. Thus, no particular rational number (fraction) has 0 probability, and the probabilities of all (infinitely many) choices still add to 1. Note that it is possible for infinitely many numbers --- in this case probabilities --- to have a finite sum. For example:
1/2 + 1/4 + 1/8 + 1/16 + ... + 1/2n + ... = 1
OK, now we come to a harder question. Instead of just considering fractions (also called rational numbers -- and remember we allow numerators to be bigger than denominators, so 1006/197 is allowable), suppose we consider all the real numbers between 1 and 10. So now we could have numbers like pi and the square-root of 2. You can think of real numbers as being decimals that go on forever. Here are some examples.
pi = 3.1415926535 8979323846 2643383280... sqrt(2) = 1. 4142135623 7309504880 1688724210 ... 7/3 = 2.3333333333 33333333333 33333333333... It turns out that many real numbers such as pi and sqrt(2) are not rational; that is, they can't be written exactly as a fraction. So, even though pi is approximately 22/7 (or, even better, 314159/100000), it is never exactly a fraction. On the other hand, fractions can always be written a decimals, but they always, eventually, repeat. For example
172/35 = 4.9142857142857142857142857... where the digit pattern 142857 repeats forever. So, among the real numbers (infinite decimals) between 1 and 10 there the those that represent rationals or fractions (i.e. those that eventually repeat) and those that aren't (i.e. infinite decimals that don't become repeating). So here is the question: You think of a real number: What's the probability that it is rational?
It turns out that there are infinitely many rationals and infinitely many irrationals (non-rationals) between 1 and 10. Does that mean that the odds are "fifty-fifty" that the number you think of is rational? To answer this, we have to find some way of determining if one of these infinities is "bigger" than the other. What could "bigger" mean?
Experts in probability measure the size of sets by using a rather complicated construct called the "Lebesgue integral" (pronounced: luh-baig' integral). According to this measure theory, the size or measure of the infinity of rationals between 1 and 10 is 0, while the measure of the collection of all reals between 1 and 10 is 9 (= 10 - 1). Thus, the measure of the irrationals must be 9 also, so that the probability of chosing an irrational number = (measure of irrationals)/(measure of all reals) = 9/9 = 1. It therefore follows that the probability of selecting a rational is 0.
It is hard to find an intuitive explanation for this. Is it saying that it is impossible to pick a rational number at random from all the numbers between 1 and 10? Or, equivalently, is it certain that any number you do pick will be irrational? Either of these assertions is clearly nonsense.
This is a good example of the necessity of asking questions correctly and understanding what the issues are. What does it mean to choose a real number (infinite decimal)? How can you specify all the infinitely many digits that go into it? The answer is: You need an algorithm or recipe to generate the digits. There is a difference between talking about all of the reals between 1 and 10 and actually choosing or constructing one of them. Some, of course, have names, like pi. Some are constructed or calculated by explicit algorithms such as sqrt(2). Rationals, being simply quotients of finite whole numbers, can simply be "written down." Once we start talking about actually constructing or choosing numbers, Lebesgue theory no longer applies and the probability of choosing a rational is by no means clearly 0. In fact, the problem of how individual reals can be constructed is so complicated that it is unclear to me whether it is even possible to come up with an answer to this question. I do know, though, that the likelihood of coming up with a rational is not 0. Just try it!
This blog is getting long, but let's return the original question: is anything impossible?
What does "impossible" mean? It could mean "self-contradictory". For example, it is impossible to find an odd number that is divisible by 2. Likewise, you can't find an isosceles triangle (two equal sides) that has three unequal angles. Basically, the definitions and logic of mathematics are set up so as to make these things impossible. But the world doesn't work with quite the precision of mathematics. When the witches tell Macbeth that "no man of woman born" can kill him, he thinks all is well until he finds that MacDuff was from his "mother's womb untimely ripped" -- i.e. he was delivered by caeserian section, not "born" in the usual sense. So we have to define our terms, and what impossible can mean in the "real" world.
Is it possible for a car to drive into a 10 meter thick concrete wall and emerge undamaged from the other side? Well, actually it is, although the probability is very very very very small. A car is made up of atoms which are in turn made up of even smaller, "subatomic" particles. By the laws of QM, a tiny enough particle can not be considered as occupying a particular region of space at a particular time, but only has a probability of being anywhere, or even somewhere, at a particular time. It isn't that we are too clumsy to measure: it is literally true that it is in many places simultaneously with different probabilities. In fact, this has been tested by experiment. A bunch of electrons are placed near an insurmountable material barrier --- the equivalent of a concrete wall. However, each has a tiny probability of actually being on the other side of that barrier. Although the probability is small, there are so many electrons that there is a real likelihood of an electron from their company appearing on the other side. An lo, in fact, electrons are actually detected on the other side of the barrier! This is a well-known effect called "tunneling" and actually has applications (try searching for "tunneling electron microscopes" to see how they work). So, it is (remotely) possible that the individual particles in a car could tunnel through and appear, fully formed, as the "same" car on the other side.
As George Gamow pointed out in his famous book "One, Two, Three...Infinity", it is possible (though fortunately highly unlikely), that all the air in your room will happen to migrate, through random collisions of air molecules, in some corner near the ceiling, leaving you suffocating in your chair. He actually computes the probability, which is tiny but non-zero, of this happening. Will also asks if it isn't possible that the probability of a coin coming up heads is not actually 1/2, since it can either come up heads, tails, land on edge or simply "float away". Yes, this is correct. Tails and Heads have pretty nearly equal likelihood, though, while landing on edge is very unlikely --- tiny probability --- and simply floating away, like the air collecting in a corner of a room, has very very very small likelihood. There are even other possibilities: the coin, a quarter, may, through some weird QM transactions, morph into two dimes and a nickel. The total of these other strange possibilities is so small, though that the likelihood of Heads (or Tails) is still very close to 1/2, as repeated experiments have shown. In fact, over this my lifetime of coin-tossing, no toss has ever resulted in anything other than Heads or Tails, though once a dime rolled under the sofa and, mysteriously, didn't surface till months later, when it reappeared as a nickel ...
