One way to build risk decay into a model is to assume that the risk is unknown within some range, and to update on survival.
A very simple version of this is to assume an unknown constant per-century extinction risk, and to start with a uniform distribution on the size of that risk. Then the probability of going extinct in the first century is 1⁄2 (by symmetry), and the probability of going extinct in the second century conditional on surviving the first is smaller than that (since the higher-risk worlds have disproportionately already gone extinct) - with these assumptions it is exactly 1⁄3. In fact these very simple assumptions match Laplace’s law of succession, and so the probability of going extinct in the nth century conditional on surviving the first n-1 is 1/(n+1), and the unconditional probability of surviving at least n centuries is also 1/(n+1).
More realistic versions could put more thought into the prior, instead of just picking something that’s mathematically convenient.
One way to build risk decay into a model is to assume that the risk is unknown within some range, and to update on survival.
A very simple version of this is to assume an unknown constant per-century extinction risk, and to start with a uniform distribution on the size of that risk. Then the probability of going extinct in the first century is 1⁄2 (by symmetry), and the probability of going extinct in the second century conditional on surviving the first is smaller than that (since the higher-risk worlds have disproportionately already gone extinct) - with these assumptions it is exactly 1⁄3. In fact these very simple assumptions match Laplace’s law of succession, and so the probability of going extinct in the nth century conditional on surviving the first n-1 is 1/(n+1), and the unconditional probability of surviving at least n centuries is also 1/(n+1).
More realistic versions could put more thought into the prior, instead of just picking something that’s mathematically convenient.