91Ó°ÊÓ

When talking about probability, an **outcome** is simply a possible result of an experiment or situation. In our example of a family with two children, each child being either a boy (B) or a girl (G), there are a certain number of possible outcomes based on the combinations of these genders.

To determine the possible outcomes when two children are born in a family, consider each individual child's gender possibility. Each child has two options: boy or girl. Therefore, for two children you can have combinations as follows:

• Boy - Boy (BB)
• Boy - Girl (BG)
• Girl - Boy (GB)
• Girl - Girl (GG)

Each of these outcomes is unique, and together they form the complete set of possibilities for this situation. Understanding these outcomes helps in further exploring how probable each one is, thereby leading into probability distribution.

Once we grasp the possible outcomes, the next step is to determine how often you might expect each outcome to occur — this is known as the **probability distribution**. For our example, we have four outcomes, each occurring with the same probability.

To find this probability, you multiply the separate probabilities of each child being a boy or a girl. Given that the probability of either child being a boy or a girl is equal, i.e., \(0.50\), calculating the probability for each listed outcome involves:

• Probability of Boy-Boy: \(0.50 \times 0.50 = 0.25\)
• Probability of Boy-Girl: \(0.50 \times 0.50 = 0.25\)
• Probability of Girl-Boy: \(0.50 \times 0.50 = 0.25\)
• Probability of Girl-Girl: \(0.50 \times 0.50 = 0.25\)

The sum of these probabilities \(0.25 + 0.25 + 0.25 + 0.25 = 1.00\) assures us that our distribution covers all possible outcomes. This balance across likely outcomes confirms that our understanding of the situation is complete and accurate.

**Combinatorics** is the branch of mathematics dealing with combinations of objects. It's an essential tool for counting possible outcomes. In our two-child family scenario, combinatorics simplifies the process of determining outcome possibilities.

In combinatorics, when calculating the number of possible outcomes, you often use the counting principle. It tells you that for every option there is another set of choices to consider. For instance, with two children and two gender possibilities per child:

• The first child can be a boy or a girl; this gives 2 possibilities.
• The second child can also be a boy or a girl, again providing 2 possibilities.

By multiplying these possibilities together, you get the total number of outcomes: \(2 \times 2 = 4\).

Thus, combinatorics facilitates understanding how different combinations arise, confirming our outcomes: Boy-Boy, Boy-Girl, Girl-Boy, and Girl-Girl. This systematic approach of counting ensures that all the variables are considered, making it a vital methodology in probability calculations.