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A random variable is a concept in probability that helps us to quantify outcomes. In this exercise, our random variable, denoted as \(x\), represents the number of lemons selected from a sample of two cars.

Random variables can be discrete or continuous. Here, \(x\) is a discrete random variable because it can take on only specific values, namely 0, 1, or 2. Each of these values represents how many of the selected cars are lemons.

By defining \(x\), we can write its probability distribution, which outlines the likelihood of each possible outcome for \(x\). This helps us understand the probabilities associated with different numbers of defective cars (lemons) from a random selection.

Probability calculation is central to determining how likely different outcomes are.

To find the probability distribution of \(x\) in this scenario, we start by calculating the likelihood of each possible value of \(x\):

• \(P(x=0)\) is the chance that both cars are not lemons. This means both cars are good, calculated as the product of their individual probabilities: \(P(GG) = 0.95 \times 0.95 = 0.9025\).


• \(P(x=1)\) accounts for exactly one car being a lemon, with two possible sequences: a good car first, followed by a lemon car or vice-versa. Hence, \(P(GL) + P(LG) = 2 \times 0.05 \times 0.95 = 0.095\).


• \(P(x=2)\) indicates both selected cars are lemons, calculated by \(P(LL) = 0.05 \times 0.05 = 0.0025\).

These values form the probability distribution of the random variable \(x\).

Tree diagrams are a wonderful visual tool in probability theory. They help us systematically map out all possible outcomes and their corresponding probabilities.

In our exercise, the tree starts at a single point (the first selection). This splits into two branches, representing the choice between a good car \(G\) and a lemon car \(L\). Each of these branches then splits again, capturing the outcome of the second selection.

Each branch is labeled with the probability of its respective event occurring. To find the probability of a complete sequence (like \(GL\) or \(LL\)), you multiply the probabilities along the branches. This structure clarifies how different combinations of events relate and ensures all possibilities are considered.

Probability theory forms the mathematical foundation for studying random events. It guides us in calculating the likelihood of different events through defined rules and formulas.

Key principles here include the multiplication rule, which helps compute the probability of sequential independent events (like our car selections). By multiplying individual event probabilities, we capture the combined likelihood of a series of events.

This foundational understanding of probability theory helps solve larger, more complex problems. It allows you to explore various configurations of events and see how likely each scenario is, fortifying your grasp of how randomness behaves in practical situations.