Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 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?

Question

Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 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?

BlackTom AI · Accepted Answer

We’re considering two families of grandparents and comparing the probability that the share of boys among their grandchildren falls between 40% and 60%, assuming equal chance of boy or girl.

Option 1: The first pair is more likely to have between 40% and 60% boys because it is likely they have exactly 50% boys which is equal to the theoretical value.
- This statement mixes up what is meant by “more likely.” For a small sample (4 grandchildren), having exactly 2 boys (which is 50%) is possible, but it is not the most probable outcome within the 40–60% range simply because 50% is just one specific outcome among several. In fact, the probability of exactly 2 boys out of 4 is C(4,2)/2^4 = 6/16 = 0.375 (37.5%). The range 40–60% includes only the 2-boy outcome for 4 grandchildren, so the event’s probability is limited to that single outcome. The statement overstates how likely it is to land exactly at 50% for such a small n, and it ignores that other outcomes within the range (if any) could occur. Overall, this option falsely asserts a higher likelihood and relies on a precise 50% that is not especially probable for n=4.

Option 2: The first pair is more likely to have between 40% and 60% boys because the second pair are having so many grandchildren it is less likely that they will have between 40% and 60% boys in accordance with the Law of Large Numbers.
- This misstates the Law of Large Numbers. The LLN says that as the number of trials grows, the observed proportion of successes tends to the true proportion (0.5 for a fair coin). It does not imply that having more grandchildren makes it less likely to fall within a given wide interval like 40–60%. In fact, it makes it more likely that the proportion is close to 50%, which lies inside that interval. Therefore, this option is incorrect because it misunderstands how LLN operates with larger samples.

Option 3: The second pair is more likely to have between 40% and 60% boys because the Law of Large Numbers suggests that the larger the number of repetitions, the closer the empirical probability of an event is likely to be to the true theoretical probability.
- This captures the correct intuition. With 31 grandchildren, the proportion of boys is a binomial proportion with p = 0.5. The LLN indicates that as n grows, the sample proportion concentrates near 0.5, so it becomes more likely that the observed proportion falls within a relatively wide interval around 0.5, such as 0.4–0.6. Therefore, the second pair has a higher chance of the proportion lying in that interval than the first pair with only 4 grandchildren, where the distribution is much more spread out and the 40–60% range is dominated by a single outcome (2 boys).

Option 4: Both pairs of grandparents are equally likely to have between 40% and 60% boys according to the Law of Large Numbers.
- This is incorrect because the LLN does not imply equal likelihood for a tiny sample and for a much larger sample. The second pair’s larger n reduces sampling variability and increases the probability that the proportion lies near 0.5, which makes it more likely to be in 40–60%, not equally likely with the small-sample first pair.

In summary, the reasoning against the first two options relies on misinterpretations: first, about exactly hitting 50% in a small sample; second, about how the Law of Large Numbers behaves with increasing sample size. The third option correctly appeals to LLN to explain why the larger-sample second pair is more likely to have a 40–60% proportion of boys, while the fourth option contradicts the LLN-style intuition by claiming equal likelihood for both sample sizes.

Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 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? 单项选择题

类似问题

Select all statements that are correct. The law of large numbers and central limit theorem (taken together) imply that:

As the sample size increases, the expected value of noise Stays the Same

Which of the following statements is true about the Law of Large Numbers?

As the number of coin flips increases, the proportion of heads should become closer to 50%.

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Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 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? 单项选择题