÷ÈÓ°Ö±²¥

The concept of sample space is fundamental in probability. It represents the set of all feasible outcomes for a given experiment or scenario. In our exercise, we are examining a family with two children. Hence, we need to list all potential gender combinations for these two children. Here, we have the following scenarios:

• Male, Male (MM)
• Male, Female (MF)
• Female, Male (FM)

In probability, the sample space is crucial as it is the foundation from which probabilities are calculated. Each combination is considered a possible outcome. Thus, understanding the complete sample space enables us to identify specific outcomes of interest later in our calculation processes.

Combinations are about selecting items from a larger set in a way where the order of items does not matter. In the context of this exercise, we're interested in the gender pairs of two children. Here, we identified three combinations: MM, MF, and FM. Each represents a distinct way to have two children with their respective genders.

It's important to note in this example that MF and FM are considered different because they represent different birth orders. While combinations help in organizing possible outcomes, they highlight how certain constraints—such as birth order or gender—can affect our final probability calculations.

Using combinations efficiently involves careful enumeration of possible outcomes and considering each unique variation. This ensures no potential situation is overlooked when assessing probabilities.

Favorable outcomes in probability refer to the specific outcomes that satisfy the condition we’re interested in. In this particular scenario, we are focused on the probability of the other child being the king's sister.

From our sample space, we identified that these outcomes are the ones where a female appears alongside the king. The gender combinations' favorable outcomes are:

• Male, Female (MF)
• Female, Male (FM)

These are the scenarios where the king, being male, has a sister.

Calculating probability involves the ratio of these favorable outcomes to the total number of outcomes in the sample space. In this case, there are 2 favorable outcomes over a total of 3 possible combinations. Consequently, the probability is computed as follows: \(\frac{2}{3}\). This represents a high likelihood, demonstrating that the probability reflects both the sample space and favored conditions.