In a family of eight children, what is the probability that (a) the third child is a girl? (b) six of the children are boys? (c) all the children are girls? (d) there are four boys and four girls? Assume that the probability of having a boy is equal to the probability of having a girl \((p=1 / 2)\).

Understanding the probability of having a boy or girl is essential when discussing genetics and family planning. In genetics, the sex of a child is determined by the X and Y chromosomes, with XX resulting in a girl and XY in a boy. Simplified assumptions consider that the chance of having either a girl or a boy is an independent event with a probability of 0.5 or 50%.

When we look at a specific child in a family, irrespective of the gender of any other children, the probability of that child being a girl is always 0.5. This constant probability holds true for each child due to the random combination of the parents' chromosomes.

The binomial distribution is a statistical tool used to represent the number of successes in a fixed number of independent trials, with only two possible outcomes, usually termed success and failure. This distribution is defined by two parameters: the number of trials (n) and the probability of success (p) in each trial.

In the context of family planning, where 'success' might denote the birth of a boy (or a girl, depending on the question), the binomial distribution provides a formula to calculate the probability of having a certain number of boys (or girls) in a specified number of children. The formula for the binomial probability is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\] where \(P(X=k)\) is the probability of k successes (e.g., k boys) in n trials (births), and \(p\) is the probability of a success in a single trial.

To perform genetic probability calculations, one must understand not only simple occurrence probabilities (such as the 50% chance of having a boy) but also how these probabilities interact across multiple events. The rules of probability, including multiplication and addition principles, play a critical role in these calculations.

The multiplication rule is used for finding the probability of the intersection of two independent events, like having a sequence of children of particular genders. For instance, the chance of having three boys consecutively is calculated by multiplying the probability of having a boy three times: \(0.5 \times 0.5 \times 0.5\). This is essential to understanding more complex genetic inheritance patterns and can be extended to a range of genetic events.

Combinatorics, the branch of mathematics concerning counting, is also instrumental when dealing with genetics, especially for calculating probabilities in larger family scenarios. Combinatorial analysis helps to define the number of ways events can occur and is crucial for understanding the binomial distribution.

For example, the binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose k successes out of n trials, which is why it appears in the binomial probability formula. So, when calculating the chance of having four boys and four girls in an eight-child family, we use combinatorics to determine that there are \( \binom{8}{4} \) ways to arrange those genders. This figure is then incorporated into the formula to calculate the overall probability.