Haigh - Probability Models

Mathematics Dept, University of Sussex, Brighton, UK

Imagine that some experiment involving random chance takes place, and use the symbol to denote the set of all possible outcomes. For example, if you throw an ordinary die, then ={1,2,3,4,5,6}. Or you might switch on the television set and ascertain what proportion of the current programme remains to be broadcast. Here would be the continuum of real numbers from zero to unity.

Consider the simple experiment of tossing a coin twice. Then you might write ={ HH , HT , TH , TT }, or, if you will simply count the number of Heads, then ={0,1,2}. In the game of Monopoly, the total score on two dice determines how far we move, and ={2,3,,12}. Ranging wider, in a dispute over the authorship of an article, would be the list of all persons who could possibly have written it. Experiment here has a very wide meaning.

An event is a collection of outcomes whose probability we wish to describe. We shall use capital letters from the front of the alphabet to denote events, so that in throwing one die we might have A ={1,2,3,4} or, equivalently, we could say A means the score is at most four. You can specify an event in any unambiguous way you like. It is enough to write B ={2,3,5} without going to the tortuous lengths to find a phrase such as B means we get a prime number. Any individual outcome is an event, but most events we shall be interested in will correspond to more than one outcome. We say that an event occurs when any of the outcomes that belong to it occur.

We can get an intuitive understanding of probability notions by thinking about a simple experiment that is repeatable indefinitely often under essentially the same conditions, such as tossing an ordinary coin. Experience tells us that the proportion of Heads will fluctuate as we repeat the experiment but, after a large number of tosses, we expect this proportion to settle down around one half. We say The probability of Heads is one half. This notion is termed the frequency interpretation of probability.

I emphasise the phrase a large number of tosses. No-one assumes the first two tosses will automatically balance out as one Head and one Tail, and the first six tosses might well all be Heads. So how many tosses are needed for the proportion of Heads to reach, and remain within, a whisker of 50 %? Try it yourself. With just ten tosses, although five Heads and five Tails are more likely than any other specific combination, as few as two or as many as eight Heads should cause no alarm bells. On the other hand, with 100 tosses, the extremes of 20 or 80 Heads would be quite astonishing we should be reasonably confident of between about 40 and 60 Heads. For 1000 tosses, the same degree of confidence attaches to the range from 470 to 530. In absolute terms, this range is wider than from 40 to 60, but, as a proportion of all the tosses, it is narrower. Were we prepared to make a million tosses, we could expect between 499000 and 501000 Heads with the same degree of confidence.

Probability is not restricted to repeatable experiments, but a different way of thinking is needed when the conditions cannot be recreated at will. Company executives make investment decisions based on their assessment of trading conditions, politicians are concerned with the probability they will win the next election. An art expert may claim to be 80 % certain that Canaletto did not paint a particular picture, a weather forecaster suggests the chance of rain tomorrow is 50 %. Here probability is being used to describe a degree of belief . To discover your own degree of belief in some event A , you could perform the following experiment, either for real, or in your imagination.

Take a flat disc, shaded entirely in black, fixed horizontally. At its centre is a pivot on which an arrow is mounted. When you spin the arrow, it will come to rest pointing in a completely random direction. A neutral observer colours a segment consisting of one quarter of the disc green, and poses the question:

If you think that A is more likely, the observer increases the size of the green segment, perhaps to one half of the disc, and poses the question again. If you think A is less likely, he similarly reduces the green region. This series of questions and adjustments continues until you cannot distinguish between the chances of the event A and the arrow settling in the green segment. Your degree of belief is then the proportion of the disc that is coloured green.

You can also use this idea to assess the probability you attach to repeatable events, such as selecting an Ace from a shuffled deck, or winning at Minesweeper or Spider Solitaire on your home computer. For the Ace problem, most people quickly agree on the figure of one in thirteen without any auxiliary aid, but different people can legitimately have quite different ideas of good answers to the other two probabilities, even if they have identical experience.

There are three fundamental laws of probability. In the ordinary meaning of language, the words impossible and certain are at the extremes of a scale that measures opinions on the likelihoods of events happening. If event A is felt to be impossible, we say that the probability of A is zero, and write P ( A )=0, whereas if B is thought sure to occur, then P ( B )=1. If you select a card at random from an ordinary deck, it is certain the suit will be either red or black, it is impossible it will be yellow. Probabilities are real numbers and, whatever the event A , its probability P ( A ) satisfies

Law 1 Always, 0 P ( A )1.

Any values of a probability outside the range from zero to one make no sense, and if you calculate a probability to be negative, or to exceed unity (you will, it happens to all of us), your answer is wrong.

The second law is there to give assurance that our list of possible outcomes really does include everything that might happen. If, for example, we write ={ H , T } when we toss a coin, we are excluding the possibility that it lands on its edge. This leads to

Law 2 P ()=1.

To motivate the final law, note first that if we are interested in event A , then we are automatically interested in the complementary event, A c , which consists precisely of those outcomes that do not belong to A . Exactly one of the events A and A c will occur, and ( A c ) c = A . Further, if we are interested in events A and B separately, it is natural to be interested in both A B and A B , respectively the ideas that at least one of A , B , and both of A , B occur. These notions extend to more than two events, so our collection of events will have three properties: