Discussion
In terms of life table functions, the chance that a child just born to a mother of age 31 and a father of age 33 will be alive till 40 years but will be orphaned by both parents is :
Answer: D
The correct expression is the one in option D, which matches the key.
The child must survive to 40, which has probability \(l_{40}/l_{0}\). During those 40 years the mother (31 at the birth) goes from age 31 to age 71 and the father from 33 to 73. The mother must die in that span, given that she was alive at 31, and the father likewise:
\[ \begin{aligned} &\frac{l_{40}}{l_{0}} \cdot \frac{l_{31} - l_{71}}{l_{31}} \\ &\quad \cdot \frac{l_{33} - l_{73}}{l_{33}} \end{aligned} \]
Option D prints the differences as \(l_{71} - l_{31}\) and \(l_{73} - l_{33}\), each of which is negative, but their product is the same as that of \((l_{31} - l_{71})(l_{33} - l_{73})\), so the value is correct. A divides by \(l_{0}\) instead of conditioning on the parents' ages, C uses \(L_{x}\) (person-years lived) and B is not a probability of this event.
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