Problem 50 - A First Course in Probability

Question

Suppose that an insurance company classifies people into one of three classes: good risks, average risks, and bad risks. The company's records indicate that the probabilities that good-, average-, and bad-risk persons will be involved in an accident over a 1 -year span are, respectively, .05, .15 and \(30 .\) If 20 percent of the population is a good risk, 50 percent an average risk, and 30 percent a bad risk, what proportion of people have accidents in a fixed year? If policyholder \(A\) had no accidents in \(2012,\) what is the probability that he or she is a good risk? is an average risk?

Step-by-Step Solution

Answer

The proportion of people having accidents in a fixed year is 17.5%. If policyholder A had no accidents in 2012, the probabilities of being in each risk class are: - Good risk: 24.24% - Average risk: 33.33% - Bad risk: 21.82%

Step 1: Calculate the Proportion of People Having Accidents in a Year

Using the Total Probability Theorem: The proportion of people having accidents in a year can be calculated as: P(Accident) = P(Accident | Good risk) × P(Good risk) + P(Accident | Average risk) × P(Average risk) + P(Accident | Bad risk) × P(Bad risk) Using the given probabilities of accidents for each risk class and the population percentages: P(Accident) = (0.05)(0.20) + (0.15)(0.50) + (0.3)(0.30)

Step 2: Calculate P(Accident) using the obtained expression

The proportion of people having accidents in a year is: P(Accident) = (0.05)(0.20) + (0.15)(0.50) + (0.3)(0.30) P(Accident) = 0.01 + 0.075 + 0.09 P(Accident) = 0.175 So, 17.5% of the population have accidents in a fixed year.

Step 3: Calculate the Probability of Being in a Risk Class Given No Accidents in a Year

Using Bayes' Theorem, the probabilities of being in each risk class given no accidents in a year are: P(Good risk | No accident) = P( No accident | Good risk) × P(Good risk) / P(No accident) P(Average risk | No accident) = P( No accident | Average risk) × P(Average risk) / P(No accident) P(Bad risk | No accident) = P( No accident | Bad risk) × P(Bad risk) / P(No accident) We already know P(Accident) from step 2. We can calculate P(No accident) as: P(No accident) = 1 - P(Accident) = 1 - 0.175 = 0.825

Step 4: Calculate the Required Probabilities of being in a Risk Class

Using the expression we obtained in Step 3, we get: P(Good risk | No accident) = (1 - 0.05) × 0.20 / 0.825 ≈ 0.2424 P(Average risk | No accident) = (1 - 0.15) × 0.50 / 0.825 ≈ 0.3333 P(Bad risk | No accident) = (1 - 0.3) × 0.30 / 0.825 ≈ 0.2182 Therefore, the probabilities of policyholder A being in each risk class given no accidents in 2012 are: - Good risk: 24.24% - Average risk: 33.33% - Bad risk: 21.82%

Key Concepts

Bayes' Theorem

In the context of insurance risk assessment, Bayes' Theorem provides us with a powerful tool for updating the probability estimates based on new information. This theorem relates the conditional and marginal probabilities of random events, allowing us to revise our beliefs when presented with evidence.

For example, if we want to know the probability that a person is a good risk, given that they had no accidents, Bayes' Theorem comes into play. We start by considering the initial probability of a person being a good risk, and how likely a good risk is to avoid accidents. Bayes' Theorem combines this information with the overall rate of accidents in the population to refine our estimate.

Specifically, the theorem states that the probability of an event B given A is equal to the likelihood of A given B times the initial likelihood of B, all divided by the likelihood of A. Mathematically, this is expressed as:
\[ P(B | A) = \frac{P(A | B) \times P(B)}{P(A)} \]
By applying this formula, we can update our beliefs about the risk category of an individual based on their accident record, and therefore make more informed insurance pricing or policy decisions.

Probability Calculation

Probability calculation is fundamental to assessing risk and making predictions in various industries, including insurance. In our exercise scenario, different risk classes (good, average, bad) have different probabilities of getting into an accident. To calculate the overall probability of an accident happening within a year, we use the Total Probability Theorem.

The Total Probability Theorem states that if you have several mutually exclusive scenarios that cover all possible outcomes, the total probability is the sum of the individual conditional probabilities of each scenario. The formula for our context looks like this:
\[ P(\text{Accident}) = P(\text{Accident | Good risk}) \times P(\text{Good risk}) + P(\text{Accident | Average risk}) \times P(\text{Average risk}) + P(\text{Accident | Bad risk}) \times P(\text{Bad risk}) \]
This allows us to understand the overall risk landscape and is crucial when determining insurance premiums and understanding the distribution of risk across a population.

Risk Assessment

Risk assessment in insurance involves quantifying the likelihood and potential cost of events like car accidents. By combining probability calculations and Bayes' Theorem, insurers can classify individuals into risk categories more accurately.

Each category is assigned a probability of filing a claim based on historical data. In our example, good risks have a lower probability of having an accident compared to bad risks. Insurers use this information to set premiums that are proportional to the level of risk, thus ensuring financial viability.

In practice, insurers collect and analyze large amounts of data on accident occurrences and other risk factors. They then apply probability calculations to this data to estimate risk levels. By performing these assessments, insurance companies strive to minimize their own risk while offering fair prices to policyholders. Such assessments are not only crucial for setting premiums but also for strategic planning and risk management within insurance companies.