Title: Structure and probability
Course: UGQuants (SSPS) > Section: Probability

Structure and probability

In describing our world in terms of variables, we will have evidence of some sort of social order when the values of these variables are not distributed randomly across cases. Here, ‘randomly’ means ‘purely by chance’. In the absence of random variation we get regular patterns. Associations between variables are evidence of regularity or structure: the fundamental basis of social order. If such patterns did not exist, social life would comprise nothing but meaningless chaos – indeed there would be no ‘social’ to life at all.

At the heart of almost all statistics is the search for such patterns or regularities - and at the heart of defining what constitutes a pattern or regularity is the concept of chance or probability. The probability of an event can range from 0 (impossible) to 1 (absolutely certain). The term probability denotes the chance or likelihood of an event occurring, or success as it is often described in statistics. The situation in which the event might or might not occur is known as a trial. Failure of the event to occur is known as... failure.

For example, we might be interested in child health and in the probability that a child would die before the age of 5 in a particular country of the world. The word 'success', of course, seems highly inappropriate here, but its meaning is that the event occurs as opposed to it failing to occur. In some developed countries the probability of a child dying before their fifth birthday is as low as .003 or .004. That is, of every 1,000 children born we can expect no more than three or four to die before the age of five. In some of the least developed sub-Saharan African countries the probability is as high as .25 or .28. In other words, we can expect around one-quarter of all children born in these countries to die before they reach their fifth birthday.

Whilst this tells us little with regard to the death or survival of any individual child, we can see that their probability of survival is not a matter of chance. Put bluntly, children born in an affluent country with good health services are far, far more likely to survive. This will also be the case if their parents are better off, they live in good housing, have access to a good diet, and so on. We can find associations between variables that give us important evidence about what kinds of factors correlate with lower infant mortality. Thus while we might have difficulty predicting the results for any individual case (see ecological fallacy), we can predict it accurately for groups of cases.

Probability (usually written as p) always refers to the proportion of trials ending in success. An 'event' can be almost anything in the context of probability: it includes, for example, answers to a survey question, or current mortgage interest rates going up, or the electoral success of a particular candidate or party. Indeed probability is relevant to almost any kind of data. The important thing is that the event we are concerned with can be thought of as taking place in the context of a trial where there are only two possibilities: either the event happens or it does not happen.

We can, therefore, think of our data in terms of the probability of it taking the form it does as opposed to other possible forms. For example, the proportions in a table, as we have already seen, can be thought of in terms of probabilities. If we find that in a group of people 43% are male, we could also express this by saying that people in this population had a .43 probability of being male. That does not mean that in the morning any member of this group of people is rather more likely to wake up as a woman than as a man, or that they will wake up as a man 43 mornings out of every 100. Rather it means that were we to pick members of this group at random we would have a .43 probability of picking a man. Were there no men in the group, the probability would be zero. Were the group all men, it would be 1.0, and so on.