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Very Old Man

Stage: 5 Challenge Level:

Li Ching-Yuen was a Chinese herbalist and longevity expert who was known to have died in 1928. He claimed to have been born in 1734, giving him a lifespan of 196 years. Investigations into birth records indicated that he was actually born in 1678, giving an even longer lifespan of 250 years!

Whilst this may seem unbelievable, is it? In this question we use statistics to look into the lifespan of very old people.

Whilst there is no conclusive historical evidence to support the birth date of Li Ching-Yuen, the following data concerning lifespans are known [at the time of writing this question (October 2008); sources given below]

• There were about 450000 people in the world aged over 100.
• There were 82 living people who were known to be over the age of 110
• There were 2 people known to be over the age of 115 (ages 115 and 116)
• There are 31 unverified claims of people over the age of 110, two of whom claimed to be aged 115 and 116.
• In the past 50 years, 25 people are known for certain to have lived beyond the age of 115.
• In the past 50 years, 2 people are known for certain to have lived beyond the age of 120 (dying at ages 120 and 122).
A hypothesis $H$ is made saying: Once you make it to your 100th birthday there is a fixed probability $p$ of surviving to your next birthday on any given subsequent birthday. For example, if $p$ were $0.05$ then the hypothesis says that on my 100th birthday there is a $5$% chance of surviving until I am $101$; on my $101$st birthday there would be a $5$% chance of surviving until I am $102$ and so on.

Does the data approximately fit this hypothesis? What values of $p$ would seem most appropriate?

Assume that the hypothesis is true with a generous value of $p=0.5$. With this hypothesis, how many 100 year olds would need to be in a room before we might feel confident that one would live to the age of 196 suggested by Li Ching-Yuen himself? How does this number compare with the number of people on earth today (6.7 billion)?

Extension: There are many statistical complications involved in predicting death rates. How many can you think of? How might these effect these statistics in future?

The data in this problem was collected from the websites of The Gerontologists and The Centenarians .

Living is a risky business. To see more about the statistics concerning living and for an estimate of your life expectance, see the Understanding Uncertainty pages.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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