With Hallowe’en memento mori all around us, we’ve been shuffling along this mortal coil and reflecting on our own mortality; well, about Actuarial Life Tables to be precise. These data sets might sound dull and lifeless (pun fully intended), but they’re a rich resource to explore when looking to extend pupils’ understanding of probability and risk.
Before exploring an Actuarial Life Table, you should ascertain your pupils’ prior conceptual understanding of probability: what are they building on? Almost certainly they will have hitherto written probabilities as the consequence of an actual or imagined experiment: dice will have been rolled, cards will have been dealt, coins will have been flicked. The probability of an outcome will have been defined as the number of times the desired outcome could (or did) occur divided by the total number of possible outcomes that could (or did) occur: for example, either imagining that if we were to flick a fair coin, or arguing that when we actually did flick the fair coin, there would be (or there were) two possible equally likely outcomes of which one, say “Heads”, is the one we want(ed), we say that “the probability of getting a Head is ½”. If we conduct an experiment investigating the probability of a drawing pin landing on its head (with the point facing upwards) and if when we drop a pin ten times it lands on its head seven times, then we express that this happened by saying that “the (experimental) probability of this drawing pin landing on its head is 0.7”. We believe that the more trials are conducted, the more consistent the ratio of “point up: point not up” landing positions will be, and so we expect that a graph of the number of trials on the horizontal axis versus the (experimentally-calculated) probability on the vertical axis will show the probability stabilizing around a consistent value as the experiment is repeated more times. Why do we believe this? What’s our justification? These ideas should be discussed with your pupils.
The coin-flicking / dice-rolling train of thought can be extended to look at survival rates in a population: we take a group of people of the same age and see how many are alive at the end of the year, and if, for example, from a group of 100 seventy year-olds, 93 are still alive on the day before their 71st birthday, then we say that the survival rate (for this group) as 0.93 (and the death rate is 0.07). If this group represents a population as a whole – and that is “a big if” that needs careful consideration of gender and both the current and former diet, environment, health, employment, housing etc. of the group’s members – we can record this result in a Life Table, the data in which are used by life insurance companies to model the probability of survival to particular ages for a member of the population.
The actuaries who work with the life tables start with the somewhat morbid concept of an imaginary cohort of 100 000 births and then they use the data collected over time from group such as our 100 seventy year-olds to predict – to model – how many will still be alive at the end of each year. For a fuller explanation of life tables, and how to read them, there’s this useful guidance from healthknowledge.org.uk.
Your pupils should pause to think about the difference between the actuaries’ reasoning and the reasoning from dice-rolling / coin-flicking. Tomorrow’s coin is plausibly the same as today’s (perhaps it’s literally the same coin), and so today’s reasoning about flicking the coin will apply plausibly when I flick the coin tomorrow: the reasoning today is projectable into the future. But, in the life table situation, tomorrow’s cause of and age at death are not the same as today’s (it’ll be a different person, for starters), so to what extent is the insurance companies’ reasoning from past data projectable? And if the reasoning is projectable, how far is it so? The fair coin of 2050 will probably behave in the same way as the fair coin of 2015 does, but the world of 2050 is likely to be very different to the world of 2015: are predictions about life and death 35 years from now at all plausible?
Your pupils will enjoy exploring a life table such as the one below, which is modelling survival (or, alternatively, death) rates in rural India in 1995-1999. The two highlighted figures show what's called the “life expectancy” of an individual: a striking 60.8 for men and 62.5 for women (click image to enlarge).
Life tables for different countries can be downloaded from lifetable.de. Your pupils can analyse the data that the tables are modelling and consider what’s the same and what’s different between different countries. What do they observe when they look at the data by gender (women’s life expectancy is four years greater than men’s in England, for example), or by age range (which periods are the riskiest in a person’s life, and why are these so?) or over time (how have the life table data for one country changed over, say, the last 20 years?)? (click image to enlarge)
If your pupils have previously considered joint probabilities for combined events, they could use the life table to model probabilities such as that of a married couple both reaching a certain age. This would be an interesting context in which to consider the independence of the events: do your pupils think that assortative mating is likely to increase or decrease the probability of a married couple both reaching a certain age? Let us know (email@example.com or on Twitter @NCETMsecondary) what they discover and how they reason about their observations.
Page header by Alan Levine (adapted), some rights reserved