Final answer:
The correct sample space for a couple having four children is option a), which represents all possible combinations of genders.
Step-by-step explanation:
The probability of having two girls and two boys is actually 3/8, which is not one of the provided options in the question. More specifically, this question involves the concept of a sample space in probability and how to determine probabilities for certain events within that sample space. The problem describes the scenario of a couple having four children, where 'b' stands for a boy and 'g' for a girl, and asks to identify the correct sample space for all the possible gender combinations of the four children.
Option a) {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg} shows the correct sample space because each letter represents the gender of a child and with four children, there should be 2^4 = 16 possible combinations, taking into account that the order matters as well since each child is distinct.
Next, the question asks for the probability of the couple having two girls and two boys. There are 6 combinations that fit this criterion (bgbg, bggb, gbbg, gbgb, bbgg, ggbb), and since there are 16 possible outcomes in total, the probability is 6/16, which simplifies to 3/8. Therefore, none of the probability options given in the question (1/4, 1/2, 3/4, 1/8) are correct. A correct approach to solving this type of problem involves listing the complete sample space and then counting the favorable outcomes that match the desired event.