Is God a Mathematician_ - Mario Livio [61]
Halley’s Life Table
The table shows, for instance, that of 710 people alive at age six, 346 were still alive at age fifty. One could then take the ratio of 346/710 or 0.49 as an estimate of the probability that a person of age six would live to be fifty. Similarly, of 242 at age sixty, 41 were alive at age eighty. The probability of making it from sixty to eighty could then be estimated to be 41/242, or about 0.17. The rationale behind this procedure is simple. It relies on past experience to determine the probability of various future events. If the sample on which the experience is predicated is sufficiently large (Halley’s table was based on a population of about 34,000), and if certain assumptions hold (such as that the mortality rate is constant over time), then the calculated probabilities are fairly reliable. Here is how Jakob Bernoulli described the same problem:
What mortal, I ask, could ascertain the number of diseases, counting all possible cases, that afflict the human body in every one of its many parts and at every age, and say how much more likely one disease is to be fatal than another…and on that basis make a prediction about the relationship between life and death in future generations?
After concluding that this and similar forecasts “depend on factors that are completely obscure, and which constantly deceive our senses by the endless complexity of their interrelationships,” Bernoulli also suggested a statistical/probabilistic approach:
There is, however, another way that will lead us to what we are looking for and enable us at least to ascertain a posteriori what we cannot determine a priori, that is, to ascertain it from the results observed in numerous similar instances. It must be assumed in this connection that, under similar conditions, the occurrence (or nonoccurrence) of an event in the future will follow the same pattern as was observed for like events in the past. For example, if we have observed that out of 300 persons of the same age and with the same constitution as a certain Titius, 200 died within ten years while the rest survived, we can with reasonable certainty conclude that there are twice as many chances that Titius also will have to pay his debt to nature within the ensuing decade as there are chances that he will live beyond that time.
Halley followed his mathematical articles on mortality with an interesting note that had more philosophical overtones. One of the passages is particularly moving:
Besides the uses mentioned in my former, it may perhaps not be an unacceptable thing to infer from the same Tables, how unjustly we repine at the shortness of our lives, and think our selves wronged if we attain not Old Age; whereas it appears hereby, that the one half of those that are born are dead in Seventeen years time, 1238 being in that time reduced to 616. So that instead of murmuring at what we call an untimely Death, we ought with Patience and unconcern to submit to that Dissolution which is the necessary Condition of our perishable Materials, and of our nice and frail Structure and Composition: And to account it as Blessing that we have survived, perhaps by many Years, that Period of Life, whereat the one half of the whole Race of Mankind does not arrive.
While the situation in much of the modern world has improved significantly compared to Halley’s sad statistics, this is unfortunately not true for all countries. In Zambia, for instance, the mortality for ages five and under in 2006 has been estimated at a staggering 182 deaths per 1,000 live births. The life expectancy in Zambia remains at a heartbreaking low of thirty-seven years.
Statistics, however, are not concerned only with death. They penetrate into every aspect of human life,