31 Probability
What is the chance of it snowing tomorrow? What is the likelihood that I will catch the
early train? What is the probability of you winning the lottery? Probability, likelihood,
chance are all words we use every day when we want to know the answers. They are
also the words of the mathematical theory of probability.
Probability theory is important. It has a bearing on uncertainty and is an
essential ingredient in evaluating risk. But how can a theory involving uncertainty
be quantified? After all, isn’t mathematics an exact science?
The real problem is to quantify probability.
Suppose we take the simplest example on the planet, the tossing of a coin.
What is the probability of getting a head? We might rush in and say the answer is
½ (sometimes expressed as 0.5 or 50%). Looking at the coin we make the
assumption it is a fair coin, which means that the chance of getting a head equals
the chance of getting a tail, and therefore the probability of a head is ½.
Situations involving coins, balls in boxes, and ‘mechanical’ examples are
relatively straightforward. There are two main theories in the assignment of
probabilities. Looking at the two-sided symmetry of the coin provides one
approach. Another is the relative frequency approach, where we conduct the
experiment a large number of times and count the number of heads. But how
large is large? It is easy to believe that the number of heads relative to the
number of tails is roughly 50:50 but it might be that this proportion would
change if we continued the experiment.
But what about coming to a sensible measure of the probability of it snowing
tomorrow? There will again be two outcomes: either it snows or it does not
snow, but it is not at all clear that they are equally likely as it was for the coin. An
evaluation of the probability of it snowing tomorrow will have to take into
account the weather conditions at the time and a host of other factors. But even
then it is not possible to pinpoint an exact number for this probability. Though
we may not come to an actual number, we can usefully ascribe a ‘degree of
belief’ that the probability will be low, medium or high. In mathematics,
probability is measured on a scale from 0 to 1. The probability of an impossible
event is 0 and a certainty is 1. A probability of 0.1 would mean a low probability