Given a family of two children (assume boys and girls equally likely, that is, probability \(\frac{1}{2}\) for each), what is the probability that both are boys? That at least one is a girl? Given that at least one is a girl, what is the probability that both are girls? Given that the first two are giris, what is the probability that an expected third child will be a boy?

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It's represented mathematically as \(P(A \, | \, B)\), which reads as 'the probability of event A occurring given that event B has occurred.' Understanding this helps us answer questions like, 'Given that at least one child is a girl, what is the probability that both children are girls?'
In our exercise, we used the fact that there are three combinations where at least one child is a girl: BG, GB, and GG. So, the probability that both children are girls, given that at least one is a girl, is: \[ P(\text{both girls} \, | \, \text{at least one is a girl}) = \frac{1}{3} \] Knowing conditional probability simplifies complex probability scenarios by focusing on specific conditions or subsets of the sample space.

Independent events are events where the occurrence or non-occurrence of one event does not affect the probability of the other. For example, the gender of the third child in our exercise is independent of the gender of the first two children. This independence allows us to say:
\[ P(\text{third child is a boy}) = \frac{1}{2} \] \The calculation above is based on the independence of each birth event, meaning previous outcomes do not change the probability of future events. This concept is essential in many areas of probability theory, as it simplifies calculations by allowing the use of multiplication for joint probabilities, such as: \[ P(\text{both children are boys}) = P(\text{first child is a boy}) \times P(\text{second child is a boy}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \] Understanding independent events is crucial for correctly assessing probabilities in scenarios where previous occurrences have no bearing on future events.

Combinatorial probability deals with calculating the likelihood of events by counting the number of ways certain outcomes can occur. This is useful in our exercise where we count different possible combinations of children's genders.
For two children, the possible gender combinations are: BB, BG, GB, and GG.
To find the probability of at least one girl: \[ P(\text{at least one girl}) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}} = \frac{3}{4} \]
Out of 4 possible combinations, 3 include at least one girl. Combinatorial methods make it easier to work with probabilities in scenarios involving multiple outcomes. Counting and listing all possible combinations provides a visual and systematic way to determine probabilities.

Probability theory is the mathematical framework for quantifying uncertainty. It equips us with tools to calculate the likelihood of various outcomes. In our exercise, we applied basic principles of probability theory to solve real-world problems.
Basic concepts involve:

• Calculating the probability of single events, e.g., a child being a boy (\( \frac{1}{2} \)).
• Combining probabilities for independent events, e.g., both children being boys (\( \frac{1}{4} \)).
• Using conditional probability to find likelihoods given specific conditions, e.g., both children being girls given at least one is a girl (\( \frac{1}{3} \)).

Understanding probability theory allows us to handle uncertainty in structured ways and make informed decisions in uncertain scenarios.