a measure of the degree of belief that an event will3.2 occur.This is the kind of definition that one finds in Bayesian books[11,19,29,30,31] and the formulation cited here is that given in the ISO Guide, of which we will talk later.
At first sight this definition does not seem to be superior to the combinatorial or the frequentistic ones. At least they give some practical rules to calculate ``something''. Defining probability as ``degree of belief'' seems too vague to be of any use. We need, then, some explanation of its meaning; a tool to evaluate it - and we will look at this tool (Bayes' theorem) later. We will end this section with some explanatory remarks on the definition, but first let us discuss the advantages of this definition.
To get a better understanding of the subjective definition of probability let us take a look at odds in betting. The higher the degree of belief that an event will occur, the higher the amount of money that someone (``a rational better'') is ready to pay in order to receive a sum of money if the event occurs. Clearly the bet must be acceptable in both directions (``coherent'' is the correct adjective), i.e. the amount of money must be smaller or equal to and not negative (who would accept such a bet?). The cases of and mean that the events are considered to be false or true, respectively, and obviously it is not worth betting on certainty. They are just limit cases, and in fact they can be treated with standard logic. It seems reasonable3.3 that the amount of money that one is willing to pay grows linearly with the degree of belief. It follows that if someone thinks that the probability of the event is , then he will bet to get if the event occurs, and to lose if it does not. It is easy to demonstrate that the condition of ``coherence'' implies that .
What has gambling to do with physics? The definition of probability through betting odds has to be considered operational, although there is no need to make a bet (with whom?) each time one presents a result. It has the important role of forcing one to make an honest assessment of the value of probability that one believes. One could replace money with other forms of gratification or penalization, like the increase or the loss of scientific reputation. Moreover, the fact that this operational procedure is not to be taken literally should not be surprising. Many physical quantities are defined in a similar way. Think, for example, of the textbook definition of the electric field, and try to use it to measure in the proximity of an electron. A nice example comes from the definition of a poisonous chemical compound: it would be lethal if ingested. Clearly it is preferable to keep this operational definition at a hypothetical level, even though it is the best definition of the concept.