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Determining outcomes is the first step in solving any probability problem. In this exercise, we need to figure out all the possible outcomes when a couple has three children. Each child can be either a boy or a girl. Since each child's gender is independent of the others, we can calculate the total number of outcomes by multiplying the possibilities for each child together. This gives us:\[ 2^3 = 8 \] This means there are 8 different possible combinations of genders for three children.
Next, we list all these possible outcomes, which are: BBB (Boy, Boy, Boy), BBG (Boy, Boy, Girl), BGB (Boy, Girl, Boy), GBB (Girl, Boy, Boy), BGG (Boy, Girl, Girl), GBG (Girl, Boy, Girl), GGB (Girl, Girl, Boy), and GGG (Girl, Girl, Girl). Understanding these steps helps in precisely determining the sample space of an experiment.

The sample space in probability is the set of all possible outcomes. For the current problem, the sample space includes all gender combinations possible when having three children.
As we listed previously, the sample space consists of: BBB, BBG, BGB, GBB, BGG, GBG, GGB, and GGG. Each of these combinations represents a unique outcome.
Visualizing the sample space is crucial as it lays the foundation for identifying favorable outcomes. It's like having a complete map before you start your journey. Having a complete list of outcomes helps clarify what results are possible before moving on to measure probabilities.

Favorable outcomes are those that meet the criteria we are interested in. In this problem, we are looking for the scenarios where at least one child is a girl.
From our sample space, the only outcome where there are no girls is BBB. Therefore, all other outcomes (BBG, BGB, GBB, BGG, GBG, GGB, GGG) are favorable because they contain at least one girl.
Recognizing favorable outcomes makes it easier to calculate the desired probability. This step narrows down the focus from all possible outcomes to only those which we are interested in for the specific question.

The fundamental formula for calculating probability is: \[ P(\text{event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
Having identified that there are 7 favorable outcomes (those with at least one girl) and knowing there are 8 possible total outcomes, we can now calculate the probability: \[ P(\text{at least one girl}) = \frac{7}{8} = 0.875 \]
This means there's an 87.5% chance that a couple with three children will have at least one girl. Through understanding the total and favorable outcomes along with the sample space, we arrive at the probability, making the process clear and logical.