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Random variables are essential in probability theory because they allow us to model and calculate unpredictable outcomes. They are used to represent numerical outcomes from random experiments or processes. In this context, choosing numbers \(x\) and \(y\) from the interval \([0,1]\) creates two random variables, typically denoted \(X\) and \(Y\). These variables are independent, meaning the selection of \(x\) does not affect the selection of \(y\) and vice versa. This independence is crucial because it simplifies how we analyze the probability of combined conditions, such as sums or maximum values, as seen in the exercise.
\(X\) and \(Y\) are uniform random variables, meaning each number in \([0,1]\) is equally likely to be chosen. This uniformity ensures an unbiased representation across the interval and simplifies the geometric probability assessments discussed in the problem.

Probability calculation is about finding the likelihood of an event occurring. In geometric probability, we calculate this by comparing areas. Each condition in the exercise can be seen as a region within the triangular space defined by \(x+y \leq 1\). For instance:

• In part (a), the condition \(|x-y|<1\) is always true within the triangle. Therefore, the probability is 1 since the entire region satisfies \(|x-y|<1\).
• For condition \(xy<\frac{1}{2}\) in part (b), the entire triangular region naturally falls below the curve \(xy=\frac{1}{2}\) because of the boundary \(x+y=1\), resulting again in a probability of 1.
• In part (c), the constraint \(\max{x,y}<\frac{1}{2}\) limits the solution to a square in the bottom-left of the region. Calculating the area ratio gives us a probability of \(\frac{1}{2}\).

Calculating these probabilities is done by measuring how much of the originally defined space actually satisfies the conditions. These geometry-based probabilities offer a visual and intuitive way to understand the likelihood of complex events.

The triangular region arises naturally from the condition \(x+y\leq 1\). This condition restricts the choices of \(x\) and \(y\) to a triangle with vertices at (0,0), (1,0), and (0,1). Hence, the area of the triangle plays a significant role in determining the probabilities of each condition.
In geometric probability, determining areas like this triangle guides our understanding of how likely different outcomes are in the context of random variables. The triangle limits the area we are interested in, setting the stage for other calculations. For example, understanding that this triangle is our base region (with an area of \(\frac{1}{2}\)) is crucial for normalizing probabilities when we calculate the probability of events within this boundary.

Analyzing the coordinate plane involves assessing the relationship between \(x\) and \(y\), particularly when they are plotted on a 2D graph. With \(x\) and \(y\) both ranging from 0 to 1, we initially start with a unit square. However, the condition \(x+y \leq 1\) confines all points to the triangular region as discussed.
Coordinate plane analysis lets us visualize and analyze regions like:

• The effectively half region defined by \(x>y\) where each half of the triangle split by the line \(x=y\) represents equal probabilities, suggesting symmetry and leading to a probability of \(\frac{1}{2}\) in step 6(e).
• Areas below certain curves or hyperbolas, such as \(xy=\frac{1}{2}\), demonstrate how random variables interact based on different constraints.

This kind of analysis is powerful as it transitions abstract conditions into tangible spaces on the coordinate plane, enabling clearer understanding and precise probability calculations.