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In the world of probability, a **random variable** is a critical concept that represents outcomes of a probabilistic event. In our example, we have the random variable \( X \), which stands for the number of cars a household owns. Random variables can be discrete or continuous. Here, \( X \) is a discrete random variable because it can only take certain values, specifically 0, 1, 2, 3, 4, or 5 cars.

Each of these values can be thought of as a possible outcome, and it carries a respective probability that indicates how likely it is to occur.

• For instance, the probability of a household owning 0 cars is 0.09.
• Similarly, owning 2 cars is more likely with a probability of 0.35.

This structured relationship between values and probabilities is where random variables come into play, serving as the foundation for constructing probability models and for further statistical analysis.

A **probability distribution** gives us a comprehensive outlook on all possible values of a random variable and their corresponding probabilities. In our case, the probability distribution for the number of cars per household is given by the probabilities associated with each value of \( X \).

By examining the distribution:

• The highest probability, 0.36, occurs at \( X = 1 \), meaning 36% of households own one car.
• The least frequent event, 0.02, is found at \( X = 5 \), indicating very few households own five cars.

This probability distribution helps us understand not just the likelihood of each outcome, but also the behavior of the overall population. For example, by summing probabilities for \( X = 3, 4, \) and 5, we interpreted that 20% of households have more cars than a two-car garage can accommodate, which aligns with real-world decisions such as designing garage space.

**Statistical analysis** involves examining, modeling, and interpreting data using various mathematical approaches. When dealing with a problem concerning the distribution of cars among households, statistical analysis is useful in answering key questions, like our initial query about garage capacity.

Here, the analysis is simplified into a few key steps:

• Identifying possible events (e.g., \( X > 2 \) means more than two cars).
• Calculating relevant probabilities (the transition from probabilities of \( X = 3, 4, 5 \) to total \( P(X > 2) \)).
• Converting the resulting probability into a percentage for greater understanding (20%).

This structured approach allows for decisions based on data and insights, such as predicting housing needs or making renovations to increase garage capacity. By leveraging probability models and conducting a thorough statistical analysis, we gain a clearer picture of the patterns within our data, informing practical, real-world choices.