Yes, if the chance of death each year is constant it turns out that remaining life expectancy is around 1/chance of death
Can you explain this is the case? Sorry if this is obvious, but I’m not getting it and can’t think offhand how to do the maths.
On population ethics, for totalists it then seems the dominating concern will be how valuable it is to have a population with longer lives, which puts the emphasis in a difference place from the value of keeping particular individuals alive longer.
It’s not necessarily obvious that this is the case.
Premise: In probability theory the chance of two independent events (which are events that don’t affect each other) happening together, like six coming up after you toss a dice and head coming up after you toss a coin, is calculated by multiplying the probability of the two events. In the case of the dice and the coin 1/6⋅1/2=1/12.
In the case of calculating expected future lifetime you need to sum all the additional number of years you could possibly live, each multiplied by their probability. This is how an expected value is calculated, and if you think about it it’s basically a weighted average: you want to know the “average” year you will live to, but in making the average each year can weight more or less depending on its probability.
It turns out, though, that in this case you can simplify the expected value formula, by only adding the probabilities of being alive at any given future year. This works, intuitively, because you are basically adding up all expected values that are made like this: 1MoreYear⋅probabilityOfSurvivingToThatYear. But what is the probability of surviving one more year? It is the probability of not dying all of the previous years! And so to find the probability you need to multiply all the independent events of not dying in a particular year between your current age and the year you are measuring the probability of arriving to. If the chance of dying any given year is constant at 1/1000 then the probability of not dying is 1−1/1000, and the multiplication is like this (1−1/1000)(1−1/1000)(1−1/1000)… the number of factors is the number of years between now and the year you are calculating the probability to arrive to. Let’s call this number k. The the multiplication becomes (1−1/1000)k
So you are basically adding up probabilities made like this: (1−1/1000)k but with k growing till infinite, since you want to account the probability of surviving to any arbitrary future year when calculating the expected value.
Why those probabilities add up exactly to 1/chance of death? I would think about it this way: when k is small, the term (1−1/1000)k is large, but still smaller than 1. The subsequent terms of the addition will be subsequently each time a little smaller. So you are basically adding up terms a little smaller than 1 but each time smaller and smaller. So what happens when you will have added 1000 terms? You will not quite reach 1000 in your sum. But this is compensated by all the subsequent super small terms you add till the infinite addition is complete.
Regarding population ethics: I finished to write the first draft of the second post in the series, and it is exactly about this topic. Can I send the Google Doc to you so you can comment in advance on it? It’s around 2k words. I know you are a moral philosopher (I remember you writing so in your post about Hippo), so it would be great to have feedback.
If you want to look up the maths elsewhere, it may be helpful to know that a constant, independent chance of death (or survival) per year is modelled by a negative binomial distribution.
Yes, looking back at this, I should have just said that on average, if someone dies with a probability of 1/1000, then he will live 999 years and die in the 1000th. And then I should have linked him the “Expectation” section of the Wikipedia page of the negative binomial distribution.