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Understanding how to calculate probabilities is essential in statistics and everyday decisions. Probabilities help us measure the likelihood that an event will occur. In our example, we are interested in finding out the probability that car repairs lead to a cost greater than \(100.
To do this, we need to use the concept of conditional probability. The idea here is to determine the chance that one event happens, given another event has already occurred. In mathematical terms, this involves using the formula for finding the intersection of two events:

• Event A: Faulty system in a car, with probability \( P(A) = 0.20 \).
• Event B: Cost of repair exceeding \)100, conditional on the car having a faulty system, with probability \( P(B|A) = 0.40 \).

The combined probability that both events A and B will occur is calculated as \( P(A \cap B) = P(A) \times P(B|A) \), which equals 0.08 or 8%.
This calculation tells us that there's an 8% chance that when a car is taken for inspection, more than $100 might be spent on fixing its pollution control system.

A faulty pollution control system is a common issue highlighted by the exercise. These systems are components in vehicles designed to manage and reduce harmful pollutants emitted from the engine. When these systems are faulty, it can lead to increased emission of toxins.
In the problem, it was established that 20% of vehicles checked have such faulty systems. This percentage, or probability \( P(A) = 0.20 \), is critical as it sets the stage for further calculations regarding repair costs.
Recognizing the probability and frequency of faulty systems helps us in assessing risks and deciding on preventive measures or actions. Ensuring these systems function properly can not only save costs but also contribute to environmental protection. Diagnosing and preparing for such issues is essential for car owners.

Repair costs can quickly add up, especially when handling specific functionalities of vehicles, like the pollution control system. The exercise focuses on instances where these costs exceed \(100.
The probability of these repair costs exceeding \)100 is conditional and depends on the car having a faulty system \( P(B|A) = 0.40 \).
This statistic tells us that whenever a car is found to have a faulty pollution control system, there is a 40% chance its repair will be expensive, that is, exceeding the $100 mark. Knowing these odds can help vehicle owners budget for and anticipate potential expenses.
Understanding repair costs and their probabilities assists not only in personal financial planning but also in broader applications such as setting industry standards and expectations for repair outcomes.

Examining event probability is crucial, especially in understanding complex situations like car repairs. Event probability simply quantifies how likely a particular outcome is to occur.
In this situation, we're looking at two interconnected events:

• A faulty pollution control system in resulting repairs.
• High costs, in particular those exceeding $100.

By calculating event probability, we are able to predict the likelihood of encountering high repair costs. Using conditional probabilities, we first identify the probability of a fault (20%), then determine the chance of expensive repairs given the fault's presence (40%).
By multiplying these probabilities, we arrive at the event probability of incurring over $100 in repair costs, which is 8%.
Thoroughly understanding event probabilities allows for better planning and decision-making, especially concerning maintenance and budgeting for car repairs.