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The Actuary The magazine of the Institute & Faculty of Actuaries

Probability: Doubting Thomas

To what extent are chance events guided by the hand of God? Many people nowadays would argue, not at all. In ancient times, however, the common opinion was the exact opposite.

The Institute’s library has recently acquired a rare and fascinating book that can be regarded as a ‘half-way house’ between the two camps — arguing that chance events are not guided by the hand of God unless, as a special exception, God himself chooses to intervene. The book starts to break down the religious barriers that had previously constituted an intellectual block to an objective consideration of chance events and had thereby prevented the emergence of a theory of probability.

The author was Thomas Gataker (1574-1654), a Church of England clergyman and a profound scholar. His book was entitled The Nature and Use of Lots and it was first published in London in 1619. The Institute has acquired the second edition of 1627, in which some of the text has been reworked to clarify and improve it.

Even today we often hear someone say, ‘I will die when my number’s up’. The outcome is predetermined by an external force and cannot be changed. It was this kind of belief that was prevalent in ancient civilisations: ‘the lots may be cast into the lap, but the issue depends wholly on the Lord’ (New English Bible, Proverbs 16.33).

People used to draw lots to determine many decisions, confident that God was directing the outcome — for example, to choose the officers of Athens and Rome, to share out booty, or to select people to be thrown overboard to lighten a sinking ship. The book gives many interesting details of the various methods used, and even of the frauds that were occasionally practised.

Gataker clearly thought deeply about the nature of chance events. He defines a chance event as a contingency or uncertainty severed from forecast and foresight. Then he points out that events which seem to happen by chance actually have a cause — for example, a fowl crossing the road or a stranger in front turning to the left or right. Devoutly, he reminds his readers of the Bible’s teaching that the providence of God extends to all things — from the alighting of a sparrow to the shedding of a hair (Matthew 10.29 and Luke 12.6).

However, he then argues that there is sometimes, but not always, a special providence in these chance events, ‘when according to God’s secret will and purpose, some special work, either of judgement or mercy, is thereby to be effected, for the glory of his name’ (or for other reasons). However, this is not true in general — there are many everyday chance events that do not show God’s will at all, and are not a work of God’s special providence. Two examples are when a hare runs out in front of someone or when every fresh glance shows a new scene coming together by chance.

In a statement of great importance, Gataker says that chance events are ‘uncertain to us because not determinable by us, though determined ordinarily by some natural cause or other’. I believe that this statement summarises much of our own thinking about probability today. He then asks how we can determine whether God’s special providence was involved or not in a chance event which has occurred. If a man travelling over Salisbury Plain finds a pitcher of water that someone has left there, having no use for it, no-one would say this was a special providence of God. If a very thirsty man, ready to die for want of water, found such a pitcher, he might think that this was due to God’s special providence, however, it would be hard to be sure that this was actually the case.

Blissful ignorance
Gataker then turns to the question of foreknowledge of an event. He says that an event, which appears to one person to have occurred by chance, may not seem so to another person who has forecast it or has foreknowledge of it. He states: ‘chance is founded, and dependeth wholly upon, Man’s ignorance’.

I believe this is a significant conclusion and one that still needs careful consideration today when considering future outcomes. Often we will carry out research before making a decision, to reduce the area of uncertainty but we are always left with a degree of ignorance. Gataker warns that effects and events are often better known than their causes. He gives an example of a servant who finds some money that he thinks was left there by chance, whereas in actual fact it was left there deliberately by his master to test his honesty. I believe it is often true of many people today, even of actuaries, that they base their work on observed effects rather than doing enough to seek out the underlying causes of what has been observed. Was this true of mortality trends 10 years ago, for example?

A general conclusion that Gataker then draws is that many events seem to us to occur by chance, even though they are foreseen by God, who decides beforehand how everything will turn out. Nearly 1700 years previously, Cicero took the opposite view, “it seems to me that God himself cannot foreknow absolutely those things which are to happen by chance and fortune. For if he knows it, then it will certainly happen; and if it will certainly happen, there is no chance in the matter.” (On Divination, translated by CD Yonge, Prometheus, 1997, page 207).

However, we must not expect God to direct chance events to fall out in a certain way so that the ‘right’ outcome occurs. To expect this is to presume more than God has promised. Thus we must not expect, for example, that the winner of an office determined by lot is fitter and more efficient than his opponent because God has determined the outcome. Nor may we conclude that the way events have turned out shows that God must have exercised his special providence.

In an interesting passage, Gataker anticipates one of the conclusions later arrived at in probability theory, namely that if there is only a small probability of an event occurring in one trial, and there are several trials, it is extremely unlikely that the event will occur in every trial: ‘Suppose that some one Minister of a whole hundred in our head City should by lot be selected to visit the pest-house [a building where plague victims were isolated — no minister would have welcomed being selected to visit it, since some strains of plague were highly infectious”, would the lot drawn in this case four or five times together… light certainly and constantly ever on the same man?’ As the same man will not be selected each time, Gataker argues that it is absurd to say that God determines the outcome, since this would imply that God is inconstant.

The book devotes a considerable amount of space to the lawfulness of games of chance and warns that we must not call on God to let us win. The conclusion is that such games are lawful provided they are played “for pleasure, not for profit; for game, not for gain”. This was a controversial statement and the book was attacked by critics (see Gataker’s life in the Oxford Dictionary of National Biography).

We may conclude that, although Gataker did not invent probability theory, he helped to lay the groundwork for it by moving the debate away from the widespread traditional belief that every chance event was the result of the will of God, to a position where this might only happen exceptionally.

This enabled mathematicians to develop a probability theory based on the assumption that each event occurred solely by chance without God’s intervention at all. The first published work on probability, De Ratiociniis in Ludo Aleae, was written by Christian Huygens (appended to F Schooten, Exercitationum Mathematicarum, Lyons, 1657, English translation published in London, 1714) and was wholly mathematical in nature.

Religious opinion
In assessing Gataker’s own contribution, it is worth remembering that religion played a much bigger part in most people’s lives than it does in western cultures today. Everyone was required by law to attend church weekly and the sermons were often lengthy, so a change in religious opinions would have filtered through to everyday life quite quickly. The book clearly made a significant impact at the time it appeared, stirring up controversy and going into a second edition. Its conclusion that games of chance were permissible in some circumstances was welcome to many people. The first English book to set out the rules of a card game appeared in 1651 — The Royall and Delightfull Game of Picquet was translated from a French work that went through several editions in the 1630s and 1640s — and probability theory was developed from a consideration of the chances in such games.

Gataker has been overlooked by most writers on the history of probability, However, Lorraine Daston, Classical Probability in the Enlightenment, Princeton, 1988, page 155, summarises his argument (though rather inadequately) as follows: “Hence Thomas Gataker and others separated profane from sacred appeals to lots, and made the distinction depend on the circumstances: the magnitude of the outcome; the intent of the participants; the overall scheme of God’s providence… ‘Lusorious’ lots became instances of God’s ordinary rather than extraordinary providence, in contrast to the traditional view that every lot ‘necessarily supposed the special providence and determining presence of God.’” [For an excellent article that recognises the importance of Gataker’s contribution, see DR Bellhouse, Probability in the Sixteenth and Seventeenth Centuries: An Analysis of Puritan Casuistry, International Statistical Review, vol.56, no.1 (April 1988), pp63-74.”

However, Gataker’s book helps to explain why probability theory was not created earlier, due to rigid religious beliefs. [For a discussion of the reasons for the failure of the ancient Greeks and Romans to develop a theory of probability, see CG Lewin, Pensions and Insurance before 1800 — a social history, Tuckwell, 2003, pp1-6.” If all chance events were directed by God, the study of how often particular outcomes occurred was not only blasphemous, but pointless, since past experience might not be a reliable guide to God’s decisions in future.

This would have been particularly true in relation to the sensitive question of when someone would die. Gataker showed how these religious beliefs could be modified credibly, in such a way that chance events could be viewed objectively for the first time. This way of thinking enabled probability theory to emerge, closely followed by the creation of actuarial science by Jan de Witt and Edmund Halley later in the 17th century. As actuaries, we perhaps owe more to Gataker than we have hitherto realised.

Chris Lewin has managed large pension schemes for much of his career. He leads the joint risk-management initiative of the actuarial and civil engineering professions, which has produced RAMP and STRATrisk.

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