Introduction:
In a population of 128 families, each with four children, we are asked to determine how many families are expected to have at least one boy and one girl. To solve this problem, we can use the concept of probability.

Assumptions:
- We assume that the probability of having a boy or a girl is equal (50%).
- We assume that the gender of each child is independent of the others.

Calculating the probability:
To calculate the probability of a family having at least one boy and one girl, we can calculate the probability of the complement event (i.e., the probability of a family having all boys or all girls) and subtract it from 1.

Probability of having all boys:
The probability of having all boys in a family with four children is (1/2)^4 = 1/16. This means that there is a 1 in 16 chance of a family having all boys.

Probability of having all girls:
Similarly, the probability of having all girls in a family with four children is also 1/16.

Probability of having at least one boy and one girl:
Now, we can calculate the probability of a family having at least one boy and one girl by subtracting the probability of having all boys or all girls from 1:

P(at least one boy and one girl) = 1 - P(all boys) - P(all girls)
P(at least one boy and one girl) = 1 - (1/16) - (1/16)
P(at least one boy and one girl) = 14/16

Calculating the expected number of families:
To calculate the expected number of families with at least one boy and one girl, we multiply the probability by the total number of families:

Expected number of families = Probability * Total number of families
Expected number of families = (14/16) * 128
Expected number of families = 112

Therefore, we can expect that out of 128 families with four children each, approximately 112 families will have at least one boy and one girl.