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Chapter 5: Problem 74

Consider two pairs of grandparents. The first pair has 4 grandchildren, and the second pair has 32 grandchildren. Which of the two pairs is more likely to have between \(40 \%\) and \(60 \%\) boys as grandchildren, assuming that boys and girls are equally likely as children? Why?

Short Answer

Expert verified
The second pair of grandparents with 32 grandchildren is more likely to have between 40% and 60% boys.

Step by step solution

01

Identify the Parameters

Firstly, identify the key parameters: number of grandchildren for Pair 1 (4) and Pair 2 (32), and the desired percentage of boys (between 40% and 60%).
02

Apply the Law of Large Numbers

The law of large numbers states that the more times you conduct a trial, the closer the outcome will be to the expected value. In this case, with boys and girls being equally likely, the expected proportion of boys would be 50%. As such, the more grandchildren there are, the more likely that the outcome will be close to this expected value.
03

Analyze the Number of Grandchildren for Each Pair

Pair 1 has fewer grandchildren (4), enabling a higher deviation from the expected 50% of boys. Pair 2, however, with a larger number of grandchildren (32), will likely produce an outcome closer to the expected 50% given the Law of Large Numbers.
04

Make a Conclusion

Therefore, the second pair of grandparents, with a larger number of grandchildren, is more likely to have between 40% and 60% boys due to the law of large numbers.

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