In Chapter 4 we showed how to create an empirical frequency distribution of the observed data. This contrasts with a theoretical probability distribution which is described by a mathematical model. When our empirical distribution approximates a particular probability distribution, we can use our theoretical knowledge of that distribution to answer questions about the data. This often requires the evaluation of probabilities.
Understanding Probability
Probability measures uncertainty; it lies at the heart of statistical theory. A probability measures the chance of a given event occurring. It is a number that takes a value from zero to one. If it is equal to zero, then the event cannot occur. If it is equal to one, then the event must occur. The probability of the complementary event (the event not occurring) is one minus the probability of the event occurring. We discuss conditional probability, the probability of an event, given that another event has occurred, in Chapter 45.
We can calculate a probability using various approaches.
- Subjective – our personal degree of belief that the event will occur (e.g. that the world will come to an end in the year 2050).
- Frequentist – the proportion of times the event would occur if we were to repeat the experiment a large number of times (e.g. the number of times we would get a ‘head’ if we tossed a fair coin 1000 times).
- A priori – this requires knowledge of the theoretical model, called the probability distribution, which describes the probabilities of all possible outcomes of the ‘experiment’. For example, genetic theory allows us to describe the probability distribution for eye colour in a baby born to a blue-eyed woman and brown-eyed man by specifying all possible genotypes of eye colour in the baby and their probabilities.
The Rules of Probability
We can use the rules of probability to add and multiply probabilities.
- The addition rule – if two events, A and B, are mutually exclusive (i.e. each event precludes the other), then the probability that either one or the other occurs is equal to the sum of their probabilities.
For example, if the probabilities that an adult patient in a particular dental practice has no missing teeth, some missing teeth or is edentulous (i.e. has no teeth) are 0.67, 0.24 and 0.09, respectively, then the probability that a patient has some teeth is 0.67 + 0.24 = 0.91.
- The multiplication rule – if two events, A and B, are independent (i.e. the occurrence of one event is not contingent on the other), then the probability that both events occur is equal to the product of the probability of each:
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