March 4, 2021

This week, we are back to the development of probability theory, which we approach through the graveyards of plague-infected 17^{th} century London.

In this series, I have written about how the initial impetus for the development of probability theory was gambling. First, Cardano explained how the probability of various outcomes of dice throws and card games could be calculated. Second, in correspondence between Pascal and Fermat, they discussed how to split the pot fairly if the game was stopped before it had finished.

At this point in history, it was becoming clear that the future was not up to God entirely, but could be predicted. The ideas were based on things like dice or card games, where all of the possible outcomes were known.

News of the concept of probability, although it was not called that at the time, spread — especially through the gambling community. As a consequence, mathematicians were peppered with questions about dice and cards. It is known that Samuel Pepys, the famous chronicler, who was also a gambler, sent a number of questions to Sir Isaac Newton about “flings of dice”. Newton responded respectfully, and correctly, but showed no further interest in the subject; he was too caught up in the development of calculus.

It is possible that the development of probability theory was hampered by the limitations of the algebra being used at the time. To work out probabilities with 18 dice, Newton would have needed to compute 6^{18}, an impossibly large number for pen and paper. Whilst he showed his workings for twelve dice (over 2 billion possible combinations), he didn't for 18 (over 100 trillion).

Working out probabilities for dice and cards is handy for gambling, where the total outcomes can be calculated, but it is not very useful in daily life when much is unknown. Enter Christiaan Huygens, a Dutch physicist and inventor of the pendulum clock. He studied the work on probability by Cardano, Galileo, Pascal and Fermat, et al. and developed it further. He also developed the concept of “expected gain”.

Expected gain, or what we now call “expectation,” is the difference between the amount you started with and the amount you finish up with if you place the same bet on each possible outcome. If you place a bet to double your money and the odds are 50-50, the “expected gain” is zero; you will win one bet and lose one.

in 1669, Huygens was the first person to take probability theory and successfully apply it to things that occurred away from the gambling tables in real life. But first, a trip across the water to London.

John Graunt was a haberdasher in 17^{th} century London. Though not particularly well educated, he started to investigate the births and deaths in England, particularly death from the bubonic plague. His purpose was to provide the authorities with a warning system for the resurgence and spread of the plague. In those days, each parish published “mortality rolls”: how many christenings and deaths there had been in the previous week or so and for deaths, what the causes were.

He collated all of this information and published it in a pamphlet in 1662, eight years after Pascal and Fermat’s correspondence. By analysing the mortality rolls, he was able to deduce the population of London and of England and Wales; experts think he came remarkably close. He also showed that death from plague was consistently underreported, hardly surprising when you consider that households would be required to immediately quarantine. And he provided the government with an estimate of the number of men in the general population of suitable age to fight in the military.

Huygens was shown Graunt’s pamphlet by his brother, who wrote to him that he, Christiaan, had another 16½ years to live. Christiaan, fascinated that life expectancy could be calculated, took Graunt’s tables and by applying probability theory calculated the probability that a person of a given age might die before a certain age. 1669: the year the life insurance industry was born.

Life insurance is purely a bet that you will die before a certain date. The insurance company knows full well, based on the data of millions of other deaths, the odds of dying. If you do die prior to the date, you win, or your relatives do; you will not be around to collect it.

It is interesting to note that Huygens used the language of lotteries to describe these calculations, as no other language was available to him. He wrote that the “number of chances that a person of 16 will die before the age of 36 equals 24 and that the number of chances that he will die after 36 equals 16, so that in a *fair game, *one should *bet *16 to 24 or 2 to 3 on the event that the person dies before 36”.

Prior to Huygens’ work, governments, and reigning kings and queens, had raised money by selling annuities on the lives of people. Without the mathematics to support the calculation, it had been a hit-and-miss affair. By applying probability theory to prior known statistics, Huygens was able to compute the odds of a person dying before a certain age, thus giving the seller of annuities an advantage in the profit department.

And now to Switzerland, where an extraordinary family lived, the Bernoullis. Three generations, eight people in all, made significant contributions to mathematics, astronomy and physics. Jakob Bernoulli, son of Nikolas, lived in Basel in the latter half of the 17^{th} century and he too was intrigued by the Graunt tables and Huygens’ work on life expectancy. The study of life expectancy was not without its detractors. Many thought that the timing of a person’s death was up to God and should not or could not be predicted.

Jakob was responsible for developing the Law of Large Numbers, a concept that is extremely important to probability theory. This law in essence says that the larger the sample of total events, the more accurately the actual probability of an event can be predicted. This then begs the question: How confident can I be based on a certain sample size? How large does the sample need to be to give me a specific level of confidence?

In 1703, when Jakob Bernoulli sent his work on life expectancy, using the Law of Large Numbers, to Leibniz, of calculus fame, Leibniz was sceptical that Bernoulli would find the answer he was looking for. “Nature has established patterns originating in the return of events, but only for the most part”. The last six words must have been like a dagger to Bernoulli’s heart. Nevertheless, Jakob carried on with his work, regardless of Leibniz’s doubt.

Suppose you have a jar and all you know is that it is filled with 5,000 balls, some red and some blue. How many would you have to draw from the jar to know, with a certain confidence level, how many red and how many blue balls are in the jar? Jakob developed his computations such that he could determine that the probability calculated for a particular sample size was within a stipulated amount of the actual probability.

Jakob Bernoulli’s work is applied in the gambling industry to this day. For one example, it allows game-testing laboratories to assess the randomness (unpredictability) of a random number generator and state with a certain level of confidence the actual house edge on the game being tested.

Jakob Bernoulli put probability theory firmly in the real world and its application to real-world problems would allow humans to start to manage risk.

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