Random experiments

Introduction

When you began to study mathematics years ago, you began with basic arithmetic in the form of understanding numbers and basic operations. Later, as your basic understanding of mathematics grew, it began to divide itself into 'pure' mathematics and 'applied' mathematics. You will note, however, that what you began with, namely arithmetic, is largely 'pure': the use of numbers to represent quantities is, in fact, very abstract, and it is only after understanding how numbers relate to quantities that you can do 'applied' mathematics by considering how arithmetic relates to 'real world' situations. For instance, taking two piles of blocks and putting them together to get a total is equivalent to taking the individual quantities and summing them together.

Just as 'mathematics' can be divided into 'pure mathematics' (i.e. theory) and 'applied mathematics' (i.e. applications), so 'statistics' can be divided into 'probability theory' and 'applied statistics'. And just as you cannot do applied mathematics without knowing any theory, you cannot do statistics without beginning with some understanding of probability theory. Furthermore, just as it is not possible to describe what arithmetic is without describing what mathematics as a whole is, it is not possible to describe what probability theory is without some understanding of what statistics as a whole is about, and in its broadest sense, it is about 'processes'.

A process is the manner in which an object changes over time. This description is, of course, both very general and very abstract, so let's look at a common example. Consider a coin. Now, the coin by itself is not a process; it is simply an object. However, if I was to put the coin through a process by picking it up and flipping it in the air, after a certain amount of time (however long it would take to land), it is brought to a final 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 the statistician is interested in. Without the process, however (i.e the object moving through time during the coin flip), there is nothing to analyze. Of course, leaving the coin stationary is also 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. Usually when 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 is very easy to understand what, precisely, probability theory is.

When we speak of probability theory as a whole, we mean the means and methods by which we quantify the possible outcomes of processes. Then, just as 'applied' mathematics takes the methods of 'pure' mathematics and applies them to real-world situations, applied statistics takes the means and methods of probability theory (i.e. the means and methods used to quantify possible outcomes of events) and applies them to real-world events in some way or another. For example, we might use probability theory to quantify the possible outcomes 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-world situation 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 up tails. 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. But here, we are already getting ahead of ourselves. So let's back up a little.