Its 6:00 AM, a beautiful day. The air is crisp, but not cold. There is a freshness that remains from the morning dew, and the smile of the sun is breaking the horizon. The door closes, your key turns the ignition switch and the current moves to your starter -but, unfortunately, the load does not turn. There is not an ignition… just the dreaded clicking… a broken car. You desperately take out the key and try again. The same events transpire, yet your desperation builds. You now open the hood and check the basic wires -everything looks right and you know its not the battery. The hour is now approaching 7:00 AM and you are late.
We have all experienced these events, and its part of the dynamics and corresponding uncertainty of the life. Our epistemic limitations and shortsighted perceptions ensure that, regardless of our best intention or most complete effort, there are outcomes we cannot control – and some of those things might, regardless of your ability, always be outside of our control.
So, to evaluate and make sense of a world bound by vast uncertainty, we use probability to quantify and organize events as subsets of the world we experience. We can think of probability as nothing more than a set of elements (we will only be considering countable sets -both finite and infinite); we can say, then, that a probability space represents our uncertainty regarding an experiment. Now, you may be wondering what a “space” is. Think of a space as a universe -which is, in our case, merely a set that can be divided up into subsets and whose points have some wiggle room such that they don’t leave the original set. Our probability space is composed of two distinct parts:
1) Sample space -this is the set that contains all the outcomes of our events (we denote this set as S -the green circle)
2) The other part is called the probability measure -which is a functions that maps the a subset of our sample space into the set of real numbers.
We shall call our function or measure P(x). We denote the set of outcomes that takes place in S as E –events (the red circle). So, P(E) informs us of what the chance that an experiment, if actualized, will produce an element that is a member of E; this “chance” is denoted by a value represented by a number in the reals.
What is most fascinating about the nature of probability is its simplicity. What I mean by “simplicity” is that we have a fairly straight-forward and directly computational process of determining most basic probabilities. Nonetheless, probability and its relation to sets does not come without some token of weirdness. For example, consider set of all real numbers. What is the probability that if I placed all the reals into some bag, that I select a rational? The probability is zero. That is, you do not have a chance of selecting a rational from the set of the reals.
But there are an infinite number of rationals! Moreover, they are dense in the reals -meaning that whatever open interval I , then Q is dense if for each | . We can also say that (that is, the closure of Q is a superset of the set it is dense in, but in this case, our set is R so the closure is equal to the “size” of R). Basically, everywhere you “look” in R, you will see at least one rational. So, why is it that the probability is zero?
First, let’s talk about size. As we have shown, there is some size to Q -Q is not small as per we can find it everywhere in R, and every irrational is an accumulation point for Q. Nonetheless, the size of R is massive. In fact, the set [0,1] has more irrational numbers than all Q. Consider, then, all Reals, that is the whole number line. The shear volume of irrationals makes the set Q look tiny if nonexistent. This is the most basic and intuitive way of thinking about the probability of selecting an element of Q as being zero. The other way deals with measure theory, and, sadly, we will not cover that here.
Getting back to the point: probability is useful, but it has some twists. It is a beautiful convention that we have formalized to help us precise and measure events that take on various states in a dynamic or chaotic nature. Nonetheless, it is merely a convention. In fact, consider the axioms of probability where the value of a probability is assigned a value in the interval [0,1]. Why [0,1]? Is there some overarching diving decree that willed this fact? Not really, it allows for simplicity and smoothness in calculation and is consistent with other perceptions.