Question

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?

Answer 1

Answer:

The second pair of grandparents have approximately 12 to 19 boys as grandchildren.

Explanation:

The probability of a grandchild being a girl or a boy is same, i.e., $\frac{1}{2}$.

The first pair of grandparents have 4 grandchildren.

The second pair of grandparents have 32 grandchildren.

If first pair of grandparents have between 40% to 60% boys as grandchildren then the number of boys are in the limit,

$(40\%\ of 4,\ 60\%\ of 4)=(1.6, 2.4)\approx(2, 3)$

Thus, the first pair of grandparents have approximately 2 to 3 boys as grandchildren.

If second pair of grandparents have between 40% to 60% boys as grandchildren then the number of boys are in the limit,

$(40\%\ of 32,\ 60\%\ of 32)=(12.8, 19.2)\approx(13, 19)$

Thus, the second pair of grandparents have approximately 12 to 19 boys as grandchildren.

So, it is clear that the second pair of grandparents are more likely to have between​ 40% and​ 60% boys as​ grandchildren.

Also according to the law of large numbers as the sample size increases the probability of an event gets closer to the theoretical probability.