Final answer:
The probability of obtaining different numbers of boys and girls in a family of seven children can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, the probability of having all boys is 1 out of 128, and the probability of having four boys and three girls is 35 out of 128.
Step-by-step explanation:
In a family of seven children, the total number of possible outcomes is 27 = 128, since each child can be either a boy or a girl, and there are two options for each child.
a. The probability of having all boys is 1 out of 128, since there is only one favorable outcome (all boys) among the 128 possible outcomes.
b. The probability of having all children of the same sex (either all boys or all girls) is 2 out of 128, since there are two favorable outcomes (all boys or all girls) among the 128 possible outcomes.
c. The probability of having six girls and one boy is 7 out of 128, since there are seven favorable outcomes (GBGGGGG, GGBGGGG, GGGBGGG, GGGGBGG, GGGGGBG, GGGGGGB, GGGGGGG) among the 128 possible outcomes.
d. The probability of having four boys and three girls is 35 out of 128, since there are 35 favorable outcomes (BBBBGGG, BBBGBGG, ... GGGGBBB, GGGGGGB) among the 128 possible outcomes.
e. The probability of having four girls and three boys is also 35 out of 128, since the number of favorable outcomes is the same as in part d.