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In the context of this probability problem, it is important to understand the set-up of the housing subdivision. This subdivision consists only of two housing styles: ranch-style homes and townhomes. Ranch-style homes are typically single-story, with a unique architectural design that often spans a larger horizontal space. Townhomes, on the other hand, are multi-story dwellings that are part of a series within a larger building structure. The problem outlines that there are twice as many townhomes as there are ranch-style homes. This means for every ranch-style home, there are two townhomes. Hence, understanding this relationship is crucial for setting up the correct equation that solves the problem. It shows the basic distribution of house types within the subdivision, confirming that the ratio of townhomes to ranch-style homes is a fundamental aspect of the entire exercise.

The problem also specifies another important condition: the distinction between townhomes with pools versus those without. This adds a layer of complexity to understanding the overall distribution of homes. According to the problem, there are three times as many townhomes with pools as there are townhomes without pools. To break it down, if you imagine the total number of townhomes without pools is represented by a variable, say \( x \), then townhomes with pools would be \( 3x \). This illustrates the necessity of knowing how these homes are categorized within the subdivision, as it directly impacts the probability calculations to follow. Understanding this separation is key to solving the probability that a given townhome will have a pool rather than not.

Calculating probability requires understanding the relationships and quantities of things you're examining. In our scenario, we need to determine the probability that a randomly selected home will be a townhome with a pool. Let's summarize: We've established that there are \( 4x \) total townhomes and this \( 4x \) includes both with and without pools. Since townhomes with pools equal \( 3x \), understanding these numbers allows us to see how many options are pool inclusive. The total number of homes is a sum of these townhomes plus the ranch-style homes:- Ranch-style homes = \( 2x \)- Total homes = \( 6x \)Thus, the probability of a townhome with a pool is calculated by dividing the number of townhomes with pools \( 3x \) by the total homes \( 6x \). This simplifies to a probability of \( \frac{1}{2} \), reflecting half of the available homes fitting this criterion.

Understanding how to deal with selecting homes within a subdivision helps form our probability model. In scenarios like these, subdivision home selection techniques encompass evaluating different types of homes and distinguishing certain properties, such as whether they have a pool. Think of it like drawing a random sample from a larger pool of choices — all while knowing the count and type of options available. By knowing the exact ratio and categories of homes, we can calculate probabilities more effectively. In this particular problem, the challenge is to consider distinct layers of selection: you pick one home, then see if it's a townhome, and if it has a pool. This step-by-step approach, where you break down the available choices by categories and layers, helps solidify foundational skills in solving GMAT probability problems. Understanding the distribution and selection criteria of homes is key to approaching similar scenarios hinging on layered selections within a specified set.