# MichaelPlant comments on A general framework for evaluating aging research. Part 1: reasoning with Longevity Escape Velocity

• 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.

• 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.

• It’s not nec­es­sar­ily ob­vi­ous that this is the case.

Premise: In prob­a­bil­ity the­ory the chance of two in­de­pen­dent events hap­pen­ing to­gether, which are events that don’t af­fect each other, 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 .

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: . 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 then the prob­a­bil­ity of not dy­ing is , and the mul­ti­pli­ca­tion is like this 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 . The the mul­ti­pli­ca­tion be­comes

So you are ba­si­cally adding up prob­a­bil­ities made like this: 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 is small, the term is large, but still smaller than . 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 but each time smaller and smaller. So what hap­pens when you will have added terms? You will not quite reach in your sum. But this is com­pen­sated by all the sub­se­quent su­per small terms you add till the in­finite ad­di­tion is com­plete.

I hope I have been clear. I don’t know if there is an eas­ier way of think­ing about it, but there prob­a­bly is. In that case I apol­o­gise, since I may be over­look­ing some re­ally ob­vi­ous piece of in­tu­ition.

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.