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When we talk about conditional probability, we are referring to the likelihood of an event occurring given that another event has already taken place. It helps us update our predictions about outcomes by factoring in new information. In our example, the condition is that one of the children, specifically the eldest, is a girl. This fact changes the scope of possible outcomes we need to consider when calculating the probability of the event 'both children are girls'.

To calculate conditional probability, we use the formula:
\[\begin{equation}P(A|B) = \frac{P(A \text{ and } B)}{P(B)}\end{equation}\]
where:

• P(A|B) represents the probability of event A given event B has occurred.
• P(A and B) refers to the probability of both A and B occurring together.
• P(B) indicates the probability of event B occurring.

In the exercise, since we know that the eldest is a girl, P(B) is 1 because that event has a 100% chance of having occurred. The 'A and B' scenario and event A both coincide since they represent the same event: both children being girls. Therefore, we do not need to calculate P(A and B) separately. This simplifies our calculation significantly.

Understanding probability outcomes is essential for grasping various probability problems. Outcomes are the possible results that can occur from a random process, and when working with probabilities, it's important to identify all possible outcomes to perform accurate calculations. In our example problem, there are initially four potential outcomes when a couple has two children: Boy-Boy (BB), Boy-Girl (BG), Girl-Boy (GB), and Girl-Girl (GG).

However, since the condition is that the eldest is a girl, we exclude the Boy-Boy (BB) and Boy-Girl (BG) outcomes. This leaves us with just two outcomes: Girl-Boy (GB) and Girl-Girl (GG). This reduction of possible scenarios is a direct application of incorporating given conditions into our probability assessment, thus simplifying the problem and guiding us towards the correct solution.

The final step in solving probability questions is the actual calculation. Probability calculation requires dividing the number of successful outcomes by the total number of possible outcomes. Using an analogy, imagine you have a bag of marbles with different colors. If you want to determine the probability of pulling out a red marble, you need to know the total number of red marbles and compare that to the overall total number of marbles in the bag.

We apply this same logic to our problem. As previously mentioned, after considering the condition that the eldest child is a girl, we have only two possible outcomes: Girl-Boy (GB) and Girl-Girl (GG). Since we're interested in the scenario where both are girls, only one outcome—Girl-Girl (GG)—is favorable. Therefore, the probability calculation is straightforward:
\[\begin{equation}Probability = \frac{1 \text{ (number of successful outcomes)}}{2 \text{ (total number of outcomes)}} = \frac{1}{2}\end{equation}\]
Understanding and executing these calculations are crucial for determining the likelihood of certain events and for making informed decisions based on probability.