# Deducing probability from limited statistical data (exercise)

In today's maths course we were given the following problem which was left as an exercise for interested students. How would I go about finding the answer to this?

According to some statistics, people aged 50–60 have a probability $p$ of living at least 15 more years. Similarly, people aged 55–65 have a probability $q$ of living at least 15 more years. What is the probability of people aged 55–60 to still be alive after that period of time?

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Assuming the probability of living 15 more years is monotonic in age, we can bound the required probability between $p$ and $q$. I don't see how you can be any more precise than that. – lazyhaze Apr 5 '12 at 16:18
@lazyhaze Obviously, $p < a < q$ or $p > a > q$ (whichever applies), where $a$ is the answer. I don't think the exercise has an exact answer, it's just "food for thought". My impressions is that $\frac{p+q}{2}$ would be a good guess, but I was wondering what other approaches there are. – Paul Manta Apr 5 '12 at 16:24
"Obviously" only given monotonicity: in the Brave New World $a$ would be higher than both $p$ and $q$ ("Youth almost unimpaired till sixty, and then, crack! the end."). $\frac{p+q}{2}$ is a good guess but probably not the best one: it interpolates linearly, while something like an exponential distribution seems more appropriate. In what type of math course were you given this exercise? – lazyhaze Apr 5 '12 at 17:03

Sometimes one sees models like this: $$\text{probability} = \frac{1}{1+e^{a\cdot(\text{age})+b}}$$ i.e. a logistic curve. But all cases I've seen of estimating $a$ and $b$ in this sort of situation are based on finitely many data points, each consisting of an age and a $0$ or a $1$ according as the event whose probability is considered happened or failed to happen in the particular case in question. You have less information than that here. Something crude like estimating the probability to be half-way between the two you were given may be the best you can do with what little information you have.