Problem 75 - A First Course in Probability

Question

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.

Step-by-Step Solution

Answer

In families with 2 children, the proportion of older sons is \( \frac{3}{4} \). In families with 3 children, the proportion of eldest sons is \( \frac{7}{8} \).

Step 1: Problem (a) - Families with 2 children

Let's analyze the problem with families in the village having 2 children. Since each child is equally likely to be a boy or a girl, there are a total of 4 different gender combinations for the 2 children. These combinations are: 1. Boy - Boy (BB) 2. Boy - Girl (BG) 3. Girl - Boy (GB) 4. Girl - Girl (GG) In the cases where there is at least one boy, there is also an older son (1st child). Thus, we can count the older sons in these scenarios. Number of older sons with 2-children families = Number of families having at least one son Using the combinations above, we have 3 valid scenarios to find the proportion of older sons: 1. BB 2. BG 3. GB So the proportion of older sons is: (3 valid combinations) / (4 total combinations) = 3/4

Step 2: Problem (b) - Families with 3 children

Now, let's analyze the problem with families in the village having 3 children. Since each child is independently and equally likely to be a boy or a girl, there are a total of 2^3=8 different gender combinations for the 3 children. These combinations are: 1. Boy - Boy - Boy (BBB) 2. Boy - Boy - Girl (BBG) 3. Boy - Girl - Boy (BGB) 4. Boy - Girl - Girl (BGG) 5. Girl - Boy - Boy (GBB) 6. Girl - Boy - Girl (GBG) 7. Girl - Girl - Boy (GGB) 8. Girl - Girl - Girl (GGG) As in the previous case, there will be an eldest son if there is at least one boy in the family. Thus, we can count the eldest sons in these scenarios: Number of eldest sons with 3-children families = Number of families having at least one son Using the combinations above, we have 7 valid scenarios to find the proportion of eldest sons: 1. BBB 2. BBG 3. BGB 12. BGG 4. GBB 54. GBG 6. GGB So the proportion of eldest sons is: (7 valid combinations) / (8 total combinations) = 7/8