Final answer:
To determine the expected number of out of 128 families with 4 children each to have at least one boy and one girl, we calculate the probability of having mixed-gender siblings by subtracting the chance of having all children of one gender from 1. The expected number of families with mixed genders is 112.
Step-by-step explanation:
To solve the problem of how many families are expected to have at least one boy and at least one girl among 128 families with four children each, we can first determine the probability of a family having only boys or only girls, then subtract this from 1 to find the probability of a mixed-gender siblings.
With the traditional probability of 1/2 for each child being a boy or a girl, the probability of a family having all boys is (1/2)^4 and the same for all girls. Adding these probabilities together gives us the likelihood of a family having all children of the same gender:
Probability(all boys) + Probability(all girls) = (1/2)^4 + (1/2)^4 = 1/16 + 1/16 = 1/8.
Now we subtract this from 1 to get the probability of at least one boy and one girl:
1 - 1/8 = 7/8.
Therefore, the expected number of families with at least one boy and one girl is:
(7/8) × 128 = 112 families.
We expect 112 families out of 128 to have both genders, based on typical gender distribution and probabilities.