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Probability is a measure of the chance that a particular event will occur. It's like predicting the likelihood of weather conditions for tomorrow, except in this case, the event is having a certain number of girls in a family of three children.
We calculate probability by dividing the number of favorable outcomes by the total number of possible outcomes in the sample space. If each outcome is equally likely, like a family having boys or girls, the calculation becomes straightforward.

• For example, the probability of having all three girls is calculated as the number of ways this can happen (which is 1) over the total number of possible gender combinations (which is 8), resulting in \( \frac{1}{8} \).

Similarly, you do the same calculation for other scenarios: no girls or exactly two girls. This approach provides a clear pathway to understanding how likely an event is to happen based on established data or patterns.

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It helps us understand how many different ways we can arrange or select something.
In this exercise, combinatorics helps us find out how many scenarios fit each condition of having different numbers of girls.

• With three children, how many ways can you have all girls? There's only one, as all have to be girls.
• To have no girls, again, there's only one way—all must be boys.
• For exactly two girls, you can choose which two children are girls in \( \binom{3}{2} = 3 \) ways. This means selecting any two positions for the girls out of the three available children.

By using combinatorics, you can confidently know you have considered every possible scenario.

A binary string is a sequence made up of just two different characters. In our problem, these characters represent "boy" and "girl" for simplicity. In computing terms, a binary string can consist of 1s and 0s.
For a family with three children, the possibilities can be seen like a binary number, with each digit representing the gender of a child. For instance, "GGG" might be translated as a string of three 1s if "G" is the binary 1.
Each position in a binary string can independently be one or the other, leading to \( 2^n \) combinations for a string of length \( n \). Here, for three children, it's \( 2^3 = 8 \) combinations. This matches our sample space, confirming our total number of potential situations from Step 1 of the solutions.

Favorable outcomes are those events or scenarios that satisfy the condition or criteria we're interested in. In this exercise, they refer to the outcomes that match the specific numbers of girls we are calculating the probability for.
To understand which outcomes are favorable, we must first determine all possible sequences of child genders (the sample space), then check which ones meet our specific conditions.

• "All girls" means only one outcome fits ("GGG").
• "No girls" is also just one outcome ("BBB").
• "Exactly two girls" can be three outcomes: "GGB", "GBG", or "BGG".

By identifying favorable outcomes, we essentially select the specific results that will be used to calculate probabilities, focusing our attention only on the events of interest and finding their likelihood.