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Probability theory is the branch of mathematics that deals with the likelihood of different outcomes in random events. It lays the foundation for predicting the behavior of systems that are governed by chance. Understanding probability is critical in a multitude of fields such as finance, insurance, and sciences. In our example, we need to identify the possible outcomes when a family has two children. The core of probability theory comes into play when we assign a probability to each of these outcomes. Considering each child can be either a boy or a girl, with an equal chance, we use the theory to determine that each of the four possible arrangements (BB, BG, GB, GG) has an equal probability, which in this case is 0.25 or 25%.

A random variable is a numerical description of the outcome of a random phenomenon. It is a variable whose value is subject to variations due to chance and, in our case, is used to represent the number of girls in a two-child family. When dealing with random variables, each possible outcome of the variable is associated with a value. For instance, we may denote the random variable as 'X' and define it in the following way: X(BB) = 0, X(BG) = 1, X(GB) = 1, and X(GG) = 2. Each of these values corresponds to the number of girls that can be born in each scenario. By setting these values, we transform the probabilistic scenarios of the family into quantifiable numbers, which allows for further mathematical analysis like calculating the expected value.

Mathematical expectation, or expected value, is a fundamental concept in probability and statistics that provides a measure of the central tendency of a random variable. To determine the expected value, one multiplies each outcome of a random variable by the probability of that outcome, and then sums all these products. This gives us the average outcome if the random scenario were to be repeated many times. In plain terms, it's the 'long-run average value' of repeated experiments or trials. For our two-child family example, the expected value of the number of girls can be calculated using the formula: \(E[X] = \text{sum}(x \times P(X = x))\). Here, x represents the possible number of girls, and P(X = x) the probability of having that number of girls. Applying our values, we get \(E[X] = 0 \times 0.25 + 1 \times 0.5 + 2 \times 0.25 = 1\). Hence, the expected number of girls in a family with two children, given that all outcomes are equally likely, is 1.