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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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    "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."](http://www.huxley.net/bnw/seven.html)). $\frac{p+q}{2}$ 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?2012-04-05

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You cannot deduce this probability, but you might be able to estimate it (in the statistician's sense of "estimate", as opposed to the mathematician's sense, which means to put upper or lower bounds on it).

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.