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When identifying a random variable, we need to consider the nature of the outcomes it represents. In this exercise, the random variable is denoted as \( x \) and it counts the number of cars that choose to stop at a specific gas station, the Texaco station.
Because it counts, it represents a specific count value for each event that occurs, making it a type of variable we can define. Random variables can be broadly categorized into two types:

• **Discrete Random Variables**: These take on a countable number of distinct values. Examples include the number of students in a class or the number of cars in a parking lot.
• **Continuous Random Variables**: These can take any value within a range. Examples include temperature, height, or time.

For the Texaco station scenario, since \( x \) counts cars (a countable quantity), it is clearly a discrete random variable.

Discrete variables, like the random variable \( x \) in our gas station example, can only take certain distinct values. We determine these values based on the given setup or conditions of the problem.
For \( x \), which counts the number of cars stopping at the Texaco station, six cars are in focus. Thus, \( x \) can take on any value between 0 and 6. These are the possible outcomes:

• 0: No cars stop at the Texaco station.
• 1: One car stops at the Texaco station.
• 2: Two cars stop at the Texaco station.
• 3: Three cars stop at the Texaco station.
• 4: Four cars stop at the Texaco station.
• 5: Five cars stop at the Texaco station.
• 6: All six cars stop at the Texaco station.

These are discrete because you can list each possible outcome individually, making them distinct and countable.

Countable outcomes are a defining characteristic of discrete random variables. In this scenario, we are interested in how many cars choose a particular station, with each number representing a potential outcome. There are seven specific outcomes possible, based on the number of cars stopping at the Texaco station.
The countability of outcomes means we can list them, just as we did with possible values (0 to 6). You can think of countable outcomes as those that you could write out in a list or count with your fingers. Here is why they matter: - They allow us to determine probabilities for each outcome directly. For instance, if stopping is a truly random event, each count from 0 to 6 can have an associated probability. - They help in structuring problems and drawing conclusions in statistical studies.
Understanding countability helps us appreciate and analyze data that can be quantified in defined steps or amounts.