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1.7.8. An example of a Dutch book is given to examine if a given person assigns subjective probabilities coherently. Consider a horse race with three horses, , and . A bookmaker offers probabilities of 1/4, 1/3, and 1/2, respectively, for these horses to win. Note that these probabilities add up to more than 1 and so violate condition 3. The corresponding odds are 3 to 1 against winning, 2 to 1 against winning and ‘evens’.

      The relationship between odds and probability is described briefly here with fuller details given in Chapter 2. An event with probability of occurring has odds of happening where . Conversely, an event that has odds of to 1 of happening has a probability of of happening and an event that has odds of to 1 of not happening has a probability of of happening. Odds of ‘evens’ correspond to or .

Suppose the odds offered by the bookmaker are accepted by the person. Thus, their beliefs do not satisfy the additivity law of probability (condition 3). If any single bet is acceptable, they can all be accepted. This is equivalent to a bet on the certain event that one of , or wins the race. The individual should therefore expect to break even on the outcome of the race; their winnings will equal their initial stake. Of course, it does not make sense to bet on the certain event as there should then be nothing to win or lose. This assumes the odds are fixed to satisfy condition 3. However, in this example, the odds do not satisfy condition 3 and the person will not break even. Suppose the following bets are placed: £3000 on to win, £4000 on to win, and £6000 on to win; i.e. £13 000 in total. If wins the bookmaker pays out £12 000, the original £3000 bet and another £9000 in accordance with the odds of 3 to 1 against. If wins, the bookmaker also pays out £12 000, the original £4000 bet and another £8000 in accordance with the odds of 2 to 1 against. If wins the bookmaker again pays out £12 000, the original £6000 bet and another £6000 in accordance with the odds of evens. Regardless of which horse has won the race, the individual has paid out £13 000 and receives £12 000 in winnings, thus incurring a loss of £1000. This situation is known as a Dutch book. The odds quoted did not satisfy condition 3. Conversely, if the set of odds determine probabilities that add up to less than 1, then the bookmaker will lose money. It would be incoherent for such odds to be set.

      Judgements are required in all aspect of scientific investigation. The elicitation of probability distributions for uncertain quantities represents a challenging work for scientists and decision‐makers. O'Hagan (2019) recently wrote:

       Subjective expert judgments play a part in all areas of scientific activity, and should be made with the care, rigour, and honesty that science demands. (p. 80)

      A discussion can be found in Section 1.7.7.

      1.7.7 Probabilities and Frequencies: The Role of Exchangeability

It is not uncommon for subjective (or personal) probabilities to be considered as a synonym for arbitrariness. This is not so; the use of subjectivism does not mean the use of acquired knowledge that is often available for consideration of relative frequencies is neglected. The main source of misunderstanding is concerned with the relationship between frequencies and beliefs. The two terms are, unfortunately, often regarded as equivalent since frequency data can be used to inform probabilities (Lindley 1991) but they are not equivalent. Dawid and Galavotti (2009, p. 100) quoted de Finetti's view:

       every probability evaluation essentially depends on two components: (1) the objective component, consisting of the evidence of known data and facts; and (2) the subjective component, consisting of the opinion concerning unknown facts based on known evidence.

      As emphasised more recently by D'Agostini (2016)

       It is a matter of fact that relative frequency and probability are somehow connected within probability theory, without the need for identifying the two concepts. (p. 13)

It is reasonable to use relative frequencies to inform measures of belief and the relationship takes the form of a mathematical theorem, de Finetti's Representation theorem. According to the theorem, the convergence of one's personal probability towards the value of observed frequencies, as the number of observations increases, is a logical consequence of Bayes' theorem if a condition called exchangeability is satisfied by the degrees of belief prior to the observations (Dawid 2004).

As an illustration of the connection between frequency and probability, consider again an urn containing a certain number of balls, indistinguishable except by their colour, which is either white or black, and the number of balls of each colour being known. The extraction of a ball from this urn defines an experiment having two and only two possible outcomes that are generally denoted as success (say, the withdrawal of a white ball) or failure (say, the withdrawal of a black ball). Let denote the event ‘a white ball is extracted’. Under the circumstances that balls are all indistinguishable from each other except for the colour, the subjective probability to extract a white ball can be assessed as the known proportion of white balls, that is, . Assuming the urn contains a large number of balls, so that the extraction of a few balls does not alter its composition substantially, individual draws (i.e. sampling⁹) will be considered as with replacement and the probability of extracting a white ball at subsequent withdrawals will still be , independently on previous observations. In this way one realises a series of so‐called Bernoulli trials (Section A.2.1), where the outcome of each trial has a constant probability independent from previous outcomes.

Suppose now the observer does not know the absolute value of balls present, nor the proportion that are of each colour. De Finetti (1931a) showed that every series of experiments having two and only two possible outcomes that can be taken as exchangeable (i.e. the probability assigned to the outcomes

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