Probability: part 1

Introduction

We can divide mathematics into pure and applied maths. Pure maths is the theory of maths and is very abstract. The work you havecovered on algebra is mostly pure maths. Applied maths is all about taking the theory (or pure maths) and applying it to the real world. To be able to doapplied maths, you first have to learn pure maths.

But what does this have to do with probability? Well, just as mathematics can be divided into pure maths and applied maths, sostatistics can be divided into probability theory and applied statistics. And just as you cannot do applied mathematics without knowing any theory, you cannotdo statistics without beginning with some understanding of probability theory. Furthermore, just as it is not possible to describe what arithmetic is withoutdescribing what mathematics as a whole is, it is not possible to describe what probability theory is without some understanding of what statistics as a wholeis about, and statistics, in its broadest sense, is about 'processes'.

A process is how an object changes over time. For example, consider a coin. Now, the coin by itself is not a process; it is simply an object. However, if I was to flip the coin (i.e. putting it through a process), after a certain amount of time (however long it would take to land), it is brought to afinal state. We usually refer to this final state as 'heads' or 'tails' based on which side of the coin landed face up, and it is the 'heads' or 'tails' that thestatistician (person who studies statistics) is interested in. Without the process there is nothing to examine. Of course, leaving the coin stationary isalso a process, but we already know that its final state is going to be the same as its original state, so it is not a particularly interesting process. Usuallywhen we speak of a process, we mean one where the outcome is not yet known, otherwise there is no real point in analyzing it. With this understanding, it isvery easy to understand what, precisely, probability theory is.

When we speak of probability theory as a whole, we mean the way in which we quantify the possible outcomes of processes. Then, just as 'applied' mathematics takes the methods of 'pure' mathematics and appliesthem to real-world situations, applied statistics takes the means and methods of probability theory (i.e. the means and methods used to quantify possibleoutcomes of events) and applies them to real-world events in some way or another. For example, we might use probability theory to quantify the possibleoutcomes of the coin-flip above as having a 50% chance of coming up heads and a 50% chance of coming up tails, and then use statistics to apply it a real-worldsituation by saying that of six coins sitting on a table, the most likely scenario is that three coins will come up heads and three coins will come uptails. This, of course, may not happen, but if we were only able to bet on ONE outcome, we would probably bet on that because it is the most probable. Buthere, we are already getting ahead of ourselves. So let's back up a little.