Question

Twenty percent of cars that are inspected have faulty pollution control systems. The cost of repairing a pollution control system exceeds $$\$ 100$$ about \(40 \%\) of the time. When a driver takes her car in for inspection, what's the probability that she will end up paying more than $$\$ 100$$ to repair the pollution control system?

Short answer

There's an 8% chance of exceeding $100 for repairs.

Step-by-step solution

Understanding the Problem

We need to determine the probability that a car will require a repair costing more than $100, given that the car is inspected. This involves two probabilities: the probability of having a faulty pollution control system and the probability that repairing it costs more than $100.

Define Given Probabilities

We know that the probability of a car having a faulty pollution control system is \( P(A) = 0.20 \) and the probability that the repair cost exceeds $100 given that it is faulty is \( P(B|A) = 0.40 \).

Use the Multiplication Rule

The probability that both events happen (i.e., the car has a faulty system and the repair cost exceeds $100) is the product of the probability of each event: \( P(A \text{ and } B) = P(A) \times P(B|A) \). Thus, \( P(A \text{ and } B) = 0.20 \times 0.40 \).

Calculate the Probability

Multiply the probabilities obtained: \( P(A \text{ and } B) = 0.20 \times 0.40 = 0.08 \). Thus, there's an 8% chance that a car inspection will lead to a cost of more than $100.

Key concepts

Conditional Probability

Conditional probability is the likelihood of an event happening given another event has already occurred. It's a fundamental concept in probability theory that helps us measure how one event can affect the probability of another event. For example, in our scenario, we want to know the chance that a repair costs over \(100, assuming the car already has a faulty pollution control system. Here, the conditional probability is given as \( P(B|A) = 0.40 \). This means there's a 40% chance that the repair will exceed \)100, provided the pollution control system is faulty.

To determine conditional probabilities, follow these steps:

• Identify the primary event (Event A), the situation that's given or assumed to be true.
• Identify the secondary event (Event B), the event whose probability is affected by the primary event.
• Use the formula: \( P(B|A) = \frac{P(A \text{ and } B)}{P(A)} \), where \( P(A \text{ and } B) \) is the probability that both events occur together.

Understanding conditional probability helps untangle complex situations by focusing on events' relationships.

Multiplication Rule

The multiplication rule allows us to find the probability of two or more events happening at the same time. It's particularly useful when dealing with conditional probabilities, as it combines the probability of one event with the conditional probability of another. In simple terms, the rule tells us that the probability of events A and B both occurring is the product of the probability of A and the probability of B given A.

In our car repair example, we have:

• The probability that a car has a faulty pollution control system (\(P(A) = 0.20\))
• The conditional probability that repairs cost more than \(100 if the system is faulty (\(P(B|A) = 0.40\))

Using the multiplication rule:

• \( P(A \text{ and } B) = P(A) \times P(B|A) \)
• Calculate: \( P(A \text{ and } B) = 0.20 \times 0.40 = 0.08 \)

Thus, the chance that a car inspection leads to costs exceeding \)100 is 8%. This method effectively assesses combined event probabilities.

Faulty System Probability

Faulty system probability is a specific type of probability that quantifies the likelihood of an error or defect in a system. In our example, it refers to cars having faulty pollution control systems. This probability is crucial because it forms the basis for further calculations involving costs or additional inspections needed.
Understanding and calculating the faulty system probability involve:

• Recognizing the proportion of total cases that can be faulty, given by \( P(A) = 0.20 \). This tells us that 20% of cars are likely to have a faulty system.
• Using this base probability to anticipate defects' potential impact on costs and maintenance.

By assessing the faulty system probability, companies, and individuals can make informed decisions about inspections and necessary repairs, optimizing costs and resources.