Probability and Life Contingencies in the Mid-seventeenth Century

Probability emerged as a discipline of applied mathematics in the second half of the seventeenth century. At this time, probability was focused on combinatorial problems and was usually applied to games of chance. The general objective was to derive probabilities for the possible outcomes of well-defined games where the underlying probability process was known (for example, the distribution of the sum of n fair dice). Statistical inference is the inverse problem—that is, the estimation of the characteristics of an unknown population based on a limited sample of n data points. Statistical inference followed as an application of probability but, as we shall see later, the transition from probability to statistics involved many mathematical and conceptual difficulties, and only fully emerged in second half of the eighteenth century.

Some antecedent discussions of probabilistic thought can be found in historical literature stretching back at least as far as the ancient Greeks, where gambling was an active pastime. There is some evidence of dice-based gambling games as far back as ancient Egypt. These games prompted some intellectuals to make probing observations in the field of randomness and frequency, but no probabilistic thinking from those days contains any significant mathematical content. There was nothing remotely close to a probabilistic equivalent of Euclid’s Elements. The reasons for this are the subject of much historical and philosophical debate.[1]

We have to go forward another 2,000 years to the thinkers of the Scientific Revolution of the sixteenth century before important precursors to modern probabilistic thinking can be identified. Cardano and Galileo provide two such examples. Gerolamo Cardano, a Dutch physician of high repute, wrote Liber de ludo alae, a significant book on the application of elementary combinatorial calculations to games of chance in the middle of the sixteenth century. The book analysed such problems as the probability distribution of the sum of two and three dice throws, enumerating the results of the various equally possible combinations of throws and hence outcomes for the sums. It was not published, however, until 1663, 100 years after it was written. Therefore, as its ideas had already been superseded by the time it was published, it was never influential in the early development of probabilistic thinking.

Galileo Galilei was the first of the great mathematicians and scientists who contributed to the development of probability, though his interest was no more than passing. Like Cardano, he considered the distribution of the totals of the throw of a number of dice. Apparently a friend or student noted that three dice can total each of nine or ten through six distinct combinations.[2] Yet gamblers viewed a total of ten as more likely than nine. Galileo tabulated all 216 possible results of the throwing of three dice and showed that a total of ten would be achieved in 27 of the cases whilst nine would arise in 25 cases. QED. This is an interesting anecdote in a couple of ways. Firstly, it highlights how tricky raw combinatorial logic can be to the human mind (the friend was correct to say each total could be achieved by six different distinct combinations of three dice scores, but he didn’t allow for the fact that these six combinations were not equally likely to occur—a throw of 4, 3, 3 occurs in three different ways (i.e. 4, 3, 3; 3, 4, 3; 3, 3, 4), whereas a throw of 3, 3, 3 can only occur one way. Secondly, it highlights that small differences in probability could be discerned through empirical observation (27/216 versus 25/216), even though no systematic records of empirical frequencies are known to have been made.

Prior to the 1650s, probabilistic thinking was limited to some elementary combinatorial analysis of games of chance such as those of Cardano and Galileo—probability at this time might be considered ‘pre-mathematical’. Conceptually, it was confined to calculating the ratio of favourable events to a total number of equally possible events in well-defined circumstances. So what of the practice of life contingencies during this period? There were two forms of life contingency in Renaissance Europe—short-term life insurance and life annuities—and their histories are quite different.

As noted above, marine insurance and, to a lesser degree, fire insurance developed into significant commercial activities much earlier than life insurance. Marine insurance was established in major European ports such as Genoa, Barcelona and Amsterdam from as early as the fourteenth century. Some short-term life insurance policies were written in England from as early as the sixteenth century, but such life insurance writing was essentially an ad hoc offshoot of the core activities of the established marine insurance underwriters. No underwriters appear to have existed in this period for the primary purpose of providing life insurance products.

Approaches to the pricing of these life insurance contracts would have been similar to those used by the underwriters in other areas of their business. Even though significant volumes of marine insurance were being written in the sixteenth century and earlier, no attempt was made to record the statistical frequency of shipwrecks until the late seventeenth century. No intellectual apparatus for statistical inference existed at this time, and even if it had done, it would arguably have been of limited use: given the variability of the political and economic environments of the times, the past may simply not have provided much useful information about the future. Standardised tables of maritime insurance prices did arise in various ports—insurers were clearly risk-sensitive—but their methods were not derived from statistical analysis of empirical data. Similarly, life insurance at the time would have been priced based on rules of thumb and with regard to the specific known characteristics of the life insured.

One of the main uses of early life insurance business was to provide protection from the risk that significant creditors died before repaying their debts. It was also used to speculate on the potential impending deaths of famous people of the time, such as the pope or monarchs. Such was its association with gambling that life insurance was made illegal in most of continental Europe (but not in England) by the end of the seventeenth century and this remained the case until the nineteenth century.

Annuities were more established than short-term life assurance as a form of life contingency contract in Europe in the Renaissance period. Whilst the speculative abuse of life insurance resulted in it being outlawed, annuities, on the other hand, were a useful instrument for borrowing in a lawful way: the riskiness of the cashflows was a crucial characteristic in circumventing the usury laws of the time. As a result, many European cities of the fifteenth, sixteenth and seventeenth centuries issued annuities as a form of municipal funding. No systematic allowance for age was made in setting these annuity prices, though in some cases individual underwriting may have been used to set the annuity price.

  • [1] See Hacking (1975) Chapter 1 for a survey of ancient probabilistic thinking and a discussion of whatmay have limited it.
  • [2] A total of ten can be produced by six distinct combinations: 6, 3, 1; 6, 2, 2; 5, 3, 2; 5, 4, 1; 4, 4, 2; 4, 3,3. A total of nine can be produced by the six distinct combinations: 6, 2, 1; 5, 3, 1; 5, 2, 2; 4, 4, 1; 4, 32; 3, 3, 3.
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