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In probability theory, a geometric distribution is a discrete distribution that models the number of trials needed for a success in repeated, independent Bernoulli trials (trials with two possible results: success or failure). In this problem, each birth is considered a trial with two outcomes: a boy (success) or a girl (failure). The probability of success, having a boy, is P(B) = 0.5 and similarly, the probability of a girl is P(G) = 0.5.

This distribution helps us understand the scenario where families keep having children until they have a boy. The number of children a family has follows a geometric distribution because the first success (boy) ends the process.

Probability is the measure of the likelihood that an event will occur. In this exercise, we are concerned with the probability of having boys and girls during childbirth. (P(B) = 0.5) represents the probability of having a boy, and (P(G) = 0.5) represents the probability of having a girl. These probabilities assume that each birth is independent of the previous ones.

Given these probabilities, each child’s birth does not affect the upcoming births, meaning that the probability remains the constant 0.5 for every single child born, irrespective of the previous birth outcomes.

Expected value is a core concept in probability theory that provides a measure of the center of a probability distribution. In the context of this exercise, the expected value helps us determine the average number of girls and boys each family will have under the king's rule.

To find the expected number of children until a boy appears, we use the properties of geometric distribution with P(B) = 0.5. The expected value (E(X)) of a geometric distribution is calculated as 1/P(B). Thus, the expected number of children is E(X) = 1/0.5 = 2. This means that, on average, each family will have 2 children. Since each child has an equal probability of being a girl, the expected number of girls in each family is 0.5 times the expected number of children, which is 0.5 * 2 = 1 girl per family.

As a result, the expected number of boys per family is also 1. Hence, the proportion of girls in each family will be 1 girl to 1 boy, or 50%, implying that the king’s decree does not change the overall gender ratio in the population.