2 - Probability

Summary

The primitive theory

According to what I will call ‘the primitive theory’, the probability that a trial or experiment will have a certain outcome is equal to the proportion of possible results of the trial which generate that outcome. There are six possible results of throwing a die, and three of these yield an even number; so the probability of that outcome is 1/2.

A major fault with this definition is that it entails incorrect attributions of probability. The chances that a biased die will show an even number may not be 1/2. In order to pre-empt this objection, one natural strategy is to require that the possible results of the trial be equally likely. The probability, according to such a modified account, would be the proportion of equally likely results which generate the outcome. However, this saves the definition from incorrect consequences only on pain of circularity. The account is now inadequate as a definition of probability since it depends upon the notion of equally likely results. In order to apply the definition in the calculation of some probability, we must already grasp what it is for the alternative results of the trial to have the same probability.

A second deficiency of the modified primitive theory is that it applies only to a restricted class of those cases in which we attribute probabilities. For example, it does not encompass the claim that the probability of getting 1 or 2 with a certain biased die is 0.154. In many cases there may be no way to divide the possible results into a set of equally probable, mutually exclusive alternatives. For this reason, the primitive theory would also have difficulty in dealing with probability claims such as:

1. (a) The probability that Oswald shot Kennedy is 0.7.

2. (b) The probability, on the basis of evidence already collected by the NASA, that there is life on Mars is very slight.

3. (c) The probability that a radium atom will decay in any given ten-second interval is 0.001.

As we have seen, the primitive theory can be rescued from circularity only by some further characterisation of equiprobability.