From a friend, an actuarial brain-teaser:
My friend Josh (55) and his wife Ericka (48) asked another couple to be their guardians if both of them die. According to actuarial tables, Josh has a 1/125 chance of dying this year and Ericka has a 1/358 chance. Without spending too much time on it, what are the chances that their friends become guardians in the next year? What about the next 9 years? (After nine years their youngest is 21).
Have at it. I’ll put up some analysis/ideas later. [-]
[Update] Here are some thoughts on the problem from my friend Jeff, a hedge fund manager near here, who originally posited the problem:
Let’s look first at the chances of them both dying in the first year. An obvious lower bound is assuming the deaths are completely independent, thus 1/125 * 1/358 =1/44750 is a good lower bound. There are two ways to proceed: try to estimate the correlation, which seems very hard, or to try to estimate the P(J|E), or the probability that Josh dies given that Ericka has died. Plane crashes are orders of magnitude less likely than car crashes, so let’s focus on car crashes. Given 36,000 automobile deaths per year, and assuming Ericka is as likely as average to die in a car crash, that means 3% of her deaths will occur in a car. The chance that Josh is a) in the car times the chance that Josh also dies is pretty small, perhaps 5-10%. So this means that the chance that Josh dies in a car crash with Ericka is about the same as both of them dying independently. There are other ways they could both die together (from a communicable disease, for instance). And, of course, the kids could die as well. So I’m going to estimate that P(J|E) is around 1/40. This means that for the first year the chances of both dying are 1/40 * 1/358 or about 1/14000. Over the next nine years the chances that each of them die almost doubles (nice estimate, Rick, right on the mark there), so we’ll make our official estimate (9*1.5)/14000 or about 1 in a 1000 that their friends become guardians.
Related posts: