Disclaimer: I am NOT a probabilist. Not only have I never taught probability, the last time I took a course in probability was in my sophomore year of college. So if this is well known (or totally wrong), forgive me.
A non-mathematician friend of mine shared this link with me. It compares the lifetime risk of dying by various means—cancer, heart disease, shark attack, etc. There are many problems with the analysis presented on this web page (for example, you are not equally likely to die from the flu in each of your 77.6 years (the average lifespan), conditional probability would be a more useful measure of risk for some of these, etc.), but I will ignore all of that. I want to focus on the last line. It says:
Lifetime risk is calculated by dividing 2003 population (290,850,005) by the number of deaths, divided by 77.6, the life expectancy of a person born in 2003.
For example, for drowning the risk is 1 in
Stated another way, they are claiming that if people die each year from a given cause, the total population is , and the life expectancy is , then the probability of dying from the given cause is . I saw this and I thought, “Surely this is wrong. Why would that formula give the probability?”
So I tried to calculate it myself. Here is my back-of-the-envelope calculation. The chance of dying from this cause in one year is . The chance of not dying from this cause in one year is , the chance of not dying from this cause for years is , and so the chance of dying from the cause in years is . (Of course, this leaves open the possibility of dying several times in those years, but we’ll ignore that.)
Let’s use this formula with the drowning example. I get , or 1 in .
What?!?! I was shocked to see an answer almost identical to the one using the “wrong” technique. There must be more to this than I first thought. Let’s look a little closer.
First, notice that . Sitting inside this expression is a sub-expression that looks a lot like the limit definition of . In particular, because is a large number, this expression is very nearly . Aha! There’s the term! But we still don’t quite have what we want.
What we’ve shown is that if the probability someone dies of a given cause in one year is , then the probability that they will die from it in years is approximately . Now suppose the probability is small (like the probability of dying by drowning). We will compute the linear approximation to this function at . We see that . At , that derivative is . So the linear approximation at is simply . In particular, if we evaluate it at our specific annual probability value , we obtain . And there it is! [Update: thank you to the commenters for pointing out that the introduction of the exponential function, while fine, is unnecessary. Quicker: just use the linear approximation for at .]
Again, I’ve never seen this before. Perhaps it is well known. For example, maybe it is a good rule-of-thumb that all good actuaries know.
I’d be happy to hear people’s thoughts about this formula and my reasoning. Maybe there’s another, different way to see this.
[I'd like to thank my colleague Jeff Forrester for talking through this with me.]