Could the birthday paradox be interpreted also about deaths?

Question

Is the probability from the birthday paradox also true about deaths? If so, why? Or why not? I would think that it is also true about deaths, but it doesn't say so.

Answer 1

Given an urn containing $n$ labeled balls, let $p_k$ be the probability that $k$ balls chosen at random (with replacement) contains a repeated ball.

The Birthday Paradox says that, when $n=365$, we have $p_k>0.5$ whenever $k \geq 23$ (the "paradox" is that $23$ seems surprisingly low).

The Birthday Paradox is often phrased in terms of birthdays since it makes it easier to communicate. It could be death-days without any mathematical problems (or any $23$ randomly chosen dates).

In practice, however, the Birthday Paradox runs into complications (e.g. twins) which will make observations unsatisfying (and presumably deaths are not entirely independent).

Answer 2

Do you mean the fact that if you select 23 people at random, there is a 50% probability that at least one pair of these people will die on the same day of the year (under the usual assumptions of only 365 days in the year and equal probability of dying on any given day).

Aside from being a bit morbid, if you select your random sample from among the living population, you may have to wait quite a while to find out whether any two of them die on the same day of the year. With birthdays you can find out very quickly if any two are the same.

But those are not mathematical concerns. The math is, in fact, the same for both problems. You can even imagine that you are rolling 23 dice that each have 365 numbered sides (and are equally likely to land on any of those sides) and you want the probability that you will roll at least one matching pair.