Question

The following questions are from ARTIST (reproduced with permission) An insurance company writes policies for a large number of newly-licensed drivers each year. Suppose \(40 \%\) of these are low-risk drivers, \(40 \%\) are moderate risk, and \(20 \%\) are high risk. The company has no way to know which group any individual driver falls in when it writes the policies. None of the low-risk drivers will have an at-fault accident in the next year, but \(10 \%\) of the moderate-risk and \(20 \%\) of the high-risk drivers will have such an accident. If a driver has an at-fault accident in the next year, what is the probability that he or she is high-risk?

Short answer

The probability is 50%.

Step-by-step solution

Understand the Problem

We need to find the probability that a driver is high-risk given they have an at-fault accident. We have three groups: low-risk (40%), moderate-risk (40%), and high-risk (20%). The probabilities of having an accident are 0% for low-risk, 10% for moderate-risk, and 20% for high-risk.

Define the Probabilities

Let's define some probabilities: \( P(L) = 0.4 \), \( P(M) = 0.4 \), and \( P(H) = 0.2 \) for the low, moderate, and high-risk groups, respectively. The probabilities of having an at-fault accident are \( P(A|L) = 0 \), \( P(A|M) = 0.1 \), and \( P(A|H) = 0.2 \).

Use Bayes’ Theorem

We want \( P(H|A) \), the probability that a driver is high-risk given they have an accident. Bayes’ Theorem gives us this: \[ P(H|A) = \frac{P(A|H) \cdot P(H)}{P(A)} \] where \( P(A) \) is the total probability of having an at-fault accident.

Find Total Probability of an Accident

To find \( P(A) \), combine the probabilities of having an accident based on each group: \[ P(A) = P(A|L) \cdot P(L) + P(A|M) \cdot P(M) + P(A|H) \cdot P(H) \]\[ P(A) = 0 \times 0.4 + 0.1 \times 0.4 + 0.2 \times 0.2 = 0.04 + 0.04 = 0.08 \]

Calculate \( P(H|A) \)

Using the total probability of an accident, compute \( P(H|A) \):\[ P(H|A) = \frac{0.2 \times 0.2}{0.08} = \frac{0.04}{0.08} = 0.5 \]

Conclusion

The probability that a driver is high-risk given they have an at-fault accident is 0.5, or 50%.

Key concepts

Understanding Conditional Probability

Conditional probability is a fundamental concept in probability theory. It is the probability of an event occurring given that another event has already occurred. Let's take an easy everyday example: consider the chance of it being sunny given that observed weather conditions predict a clear sky. This is a conditional probability.

In the case of the insurance company, we're interested in the probability that a driver is high-risk if they've had an at-fault accident. This is denoted as \( P(H|A) \), where \( H \) stands for high-risk, and \( A \) is the event "has an accident." This expression means we are interested in the proportion of high-risk drivers among those who have had an accident.

• If you know the probability of having an accident in each driver category, you can apply, as seen, Bayes' Theorem to find this conditional probability.
• Conditional probability helps make informed decisions and helps to understand complex statistical relationships in data.

Applying Risk Assessment

Risk assessment involves evaluating the likelihood and potential impact of certain risks. In this scenario, the insurance company assesses risk by categorizing drivers into three groups based on their potential to have an at-fault accident.

The three risk categories are:

• Low-risk: 0% chance of an accident.
• Moderate-risk: 10% chance of an accident.
• High-risk: 20% chance of an accident.

Understanding these risk levels is crucial for the insurance company when defining policy terms and premiums. The goal is to calculate a fair policy premium that compensates for the risk without deterring potential clients.

This kind of risk assessment ensures that companies can maintain financial stability even in unfortunate events like accidents. By correctly assessing risk, companies can also create incentives for drivers to adopt safer driving habits.

Mastering Probability Calculation with Bayes' Theorem

Probability calculation is a mathematical method to determine the likelihood of an event. Bayes' Theorem is a powerful tool used in probability theory that helps update initial beliefs about events using new data. It enables effective probability calculation when dealing with conditional probabilities.

• In our problem, the probability calculation aims to find the chance of being a high-risk driver if an accident has occurred.
• Start by finding individual probabilities such as \( P(H) \) for high-risk and \( P(A|H) \) for having an accident as a high-risk driver.

Bayes' Theorem is then applied:\[P(H|A) = \frac{P(A|H) \cdot P(H)}{P(A)}\]where:

• \( P(A) \) is the total probability of the event (having an accident) occurring in any risk group.
• Calculate \( P(A) \) by summing up the weighted probabilities of having an accident for each group.

Understanding and calculating these probabilities efficiently allows entities to make data-driven decisions in complex probabilistic scenarios.