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The introduction of subjective probability through a betting scheme is straightforward. The concept is based on hypothetical bets (Scozzafava 1987):

The force of the argument does not depend on whether or not one actually intends to bet, yet a method of evaluating probabilities making one a sure loser if he had to gamble (whether or not he really will act so) would be suspicious and unreliable for any purposes whatsoever. (p. 685)

Consider a proposition that can only take one of two values, namely, ‘true’ and ‘false’. There is a lack of information on the actual value of and an operational system is needed for the quantification of the uncertainty about imparted by the lack of information. A value is regarded as an amount to be paid to bet on with the conditions that a unit amount will be paid if is true and nothing will be paid if is false. In other words, is the amount to be paid to obtain an amount equal to the value of , that is associating the value 1 with ‘true’ and the value 0 with ‘false’. This idea was expressed by de Finetti (1940) in the following terms.

The probability of event is, according to Mr NN, equal to 0.37, meaning that if the person was forced to accept bets for and against event , on the basis of the betting ratio which he can choose as he pleases, this person would choose . (p. 113)⁸

Coherence, as briefly described in Section 1.7.2, is defined by the requirement that the choice of does not make the player a certain loser or a certain winner. Denote an event which is certain, sometimes known as a universal set, as and an event which is impossible, sometimes known as the empty set, as so that if and the two possible gains are

When or , there is no uncertainty in the outcome of the corresponding bet and so the coherence (in the absence of uncertainty) requires the respective gains to be zero. The values of the gains are therefore

This happens when for and for . Therefore if the subjective probability of , that represents our degree of belief on , is defined as an amount , which makes a personal bet on the event or proposition coherent, then the probability satisfies two conditions.

1 (1) ;

2 (2) .

Consider the case of possible bets on events that partition ; i.e. are mutually exclusive and exhaustive (Scozzafava 1987, p. 686). Let , be the amount paid for a coherent bet on . These bets can be regarded as a single bet on with amount . Another condition may be specified from the requirement of coherence, namely

1 (3) .

These conditions are the axioms of probability. Further details are given by de Finetti (1931b) and in Section 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.