Predicting the Future
The term Predicting the Future conjures up images of veiled women staring into hazy crystal balls or bearded men with darting eyes passing their hands over cups of tea leaves or something else equally humorously mysterious. We call these people fortune tellers, and relegate their professions to the regime of carnival side‐show entertainment, along with snake charmers and the like. For party entertainment, we bring out a Ouija board; everyone sits around the board in a circle and watches the board extract its mysterious energy from our hands while it answers questions about things‐to‐come.
On the one hand, we all seem to have firm ideas about the future based on consistent patterns of events that we have observed. We are pretty sure that there will be a tomorrow, and that our clocks will all run at the same rate tomorrow as they did today. If we look in the newspaper (or these days, on the Internet), we can find out what time the sun will rise and set tomorrow – and it would be difficult to find someone willing to place a bet that this information is not accurate. On the other hand, whether or not you will meet the love of your life tomorrow is not something you expect to see accurately predicted in the newspaper.
We seem willing to classify predictions of future events into categories of the knowable and the unknowable. The latter category is left to carnival fortune tellers to illuminate. The former category includes predictions of when you'll next need a haircut, how much weight you'll gain if you keep eating so much pizza, etc.
But, there does seem to be an intermediate area of knowledge of the future. Nobody knows for certain when you're going to die. An insurance company, however, seems able to consult its mystical Actuarial Tables and decide how much to charge you for a life insurance policy. How can an insurance company do this if nobody knows when you're going to die? The answer seems to lie in the fact that if you study thousands of people similar in age, health, life style, etc., you can calculate an average life span – and that if the insurance company sells enough insurance policies with rates based upon this average, in a financial sense this is as good as if the insurance company knows exactly when you are going to die. There is, therefore, a way to describe life expectancies in terms of the expected behavior of large groups of people of similar circumstances.
When predicting future events, you often find yourself in these situations. You know something about future trends but you do not know exactly what is going to happen. If you flip a coin, you know you'll get either heads or tails, but you don't know which. If you flip 100 coins, or equivalently flip one coin 100 times, however, you'd expect to get approximately 50 heads and 50 tails.
If you roll a pair of dice1 you know that you'll get some number between two and twelve, but you don't know which number you'll get. You do know that it's more likely that you'll get six than that you'll get two.
When you buy a new light bulb, you may see written on the package “estimated lifetime 10,000 hours.” You know that this light bulb might last 10 346 hours, 11 211 hours, 9587 hours, 12 094 hours, or any other number of hours. If the bulb turns out to last 11 434 hours you won't be surprised, but if it only lasts 1000 hours you'd probably switch to a different brand of light bulbs.
There is a hint in each of these examples which shows that even though you couldn't accurately predict the future, you could find some kind of pattern that teaches you something about the nature of the future. Finding these patterns, working with them, and learning what knowledge can and cannot be inferred from them is the subject matter of the study of probability and statistics.
We can separate our study into two classes of problems. The first of these classes is understanding the likelihood that something might occur. We'll need a rigorous definition of likelihood so that we can be consistent in our evaluations. With this definition in hand, we can look at problems such as “How likely is it that you can make money in a simple coin flipping game?” or “How likely is it that a certain medicine will do you more good than harm in alleviating some specific ailment?” We'll have to define and discuss random events and the patterns that these events fall into, called Probability Distribution Functions (PDFs). This study is the study of Probability.
The second class of problems involves understanding how well you really know something. We will only discuss quantifiable issues, not “does she really love me?” or “is this sculpture a fine work of art?”
The uncertainties in how well we know something can come from various sources. Let's return to the example of light bulbs. Suppose you're the manufacturer of these light bulbs. Due to variations in materials and...
