In a certain village, it is traditional for the eldest son...

In a certain village, it is traditional for the eldest son (or the older son in a two-son family) and his wife to be responsible for taking care of his parents as they age. In recent years, however, the women of this village, not wanting that responsibility, have not looked favorably upon marrying an eldest son. (a) If every family in the village has two children, what proportion of all sons are older sons? (b) If every family in the village has three children, what proportion of all sons are eldest sons? Assume that each child is, independently, equally likely to be either a boy or a girl.

In a certain village, it is traditional for the eldest son (or the older son in a two-son family) and his wife to be responsible for taking care of his parents as they age. In recent years, however, the women of this village, not wanting that responsibility, have not looked favorably upon marrying an eldest son.
(a) If every family in the village has two children, what proportion of all sons are older sons?
(b) If every family in the village has three children, what proportion of all sons are eldest sons? Assume that each child is, independently, equally likely to be either a boy or a girl.

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Key Concepts

Independent Events

In probability theory, independent events refer to occurrences that do not affect each other’s outcomes. In problems like this, the gender of each child is determined independently, which allows for straightforward computation of the likelihood of various gender sequences by multiplying the individual probabilities.

Equally Likely Outcomes and Sample Space Enumeration

This concept involves listing all possible outcomes of an experiment where each outcome is equally probable. In the context of families with multiple children, each unique sequence of genders represents an outcome in the sample space. Enumerating these allows for precise counting of cases—for example, determining in how many cases a son is the oldest or eldest among siblings.

Counting Under Conditions and Expected Value

Often in probability, especially when dealing with ordered events, it is useful to count occurrences subject to a condition—such as identifying the first male child (the eldest son). By combining this counting with the expected value (or average) of certain events per family, one can determine overall proportions in the population.