Exponentially-distributed lifetimes (death process) In a pure death process where the individual death rate is fixed at v, because the process is a time-homogeneous Markov process, the wating time till the next "event" (i.e. the wating time till the individual's death, i.e. the remaining lifetime of the individual) is exponentially distributed. By the memorylessness of the exponential distribution, the lifetime of the individual is also exponentially distributed with the same parameter v. This means that if the individual has lived 20 yrs, the probability that they are going to live 30 more years is the same as the probability of having a lifespan of 30 years. And that the expected lifetime of the individual is the same as the expected remaining lifetime. Something seems amiss with my argument above. Where did I go wrong? Thanks.

Nothing is wrong, you are rediscovering the so-called memoryless property of the exponential distribution. This is the fact that if $X$ is exponentially distributed, then, for every nonnegative $t$ and $s$, $P[X\geqslant t+s\mid X\geqslant s]=P[X\geqslant t]$ (and this probability is $\mathrm e^{-\lambda t}$, where $\lambda$ is the parameter of the exponential distribution).