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This problem deals with probability and classification based on the insurance data for three types of insured individuals: "cautious" (谨慎的), "general" (一般的), and "risky" (冒失的). Let me break down the information and solve the two questions:

### Given Information:
- The three categories of insured individuals have different probabilities of having an accident within a year:
  - "Cautious" (谨慎的): 0.05
  - "General" (一般的): 0.15
  - "Risky" (冒失的): 0.30
- The distribution of people in these categories:
  - "Cautious" (谨慎的): 20%
  - "General" (一般的): 50%
  - "Risky" (冒失的): 30%

### (1) Find the overall probability that an insured individual will have an accident within a year.

This is a weighted average of the accident probabilities for the three groups, based on their proportions in the population. The formula is:

\[
P(\text{accident}) = 0.20 \times 0.05 + 0.50 \times 0.15 + 0.30 \times 0.30
\]

Let me calculate this.The overall probability that an insured individual will have an accident within a year is 0.175, or 17.5%.

### (2) If an individual has had an accident within a year, find the probability that they belong to the "cautious" category (谨慎的).

We use **Bayes' Theorem** for this, which is expressed as:

\[
P(\text{cautious | accident}) = \frac{P(\text{accident | cautious}) \cdot P(\text{cautious})}{P(\text{accident})}
\]

Where:
- \(P(\text{accident | cautious}) = 0.05\)
- \(P(\text{cautious}) = 0.20\)
- \(P(\text{accident}) = 0.175\) (calculated in part 1)

Let's calculate this.The probability that an individual belongs to the "cautious" category, given that they have had an accident within a year, is approximately 0.057, or 5.71%.

Would you like more details on this process or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the probabilities change if the distribution of "cautious", "general", and "risky" individuals were different?
2. What is the probability that a person is from the "general" category, given they had an accident?
3. Can you use this method to determine the probability for someone being in the "risky" category after an accident?
4. How does Bayes' Theorem work for multiple categories?
5. How can we visualize these probabilities using a tree diagram?

**Tip**: When dealing with probabilities of conditional events, Bayes' Theorem is an essential tool to reverse conditions and find updated beliefs based on evidence.

An insurance company classifies insured individuals into three categories: 'cautious', 'general', and 'risky'. The probabilities of having an accident within a year for each category are 0.05, 0.15, and 0.30, respectively. The proportion of insured individuals in these categories are: 'cautious' 20%, 'general' 50%, and 'risky' 30%. The two tasks are: (1) Find the overall probability of an insured individual having an accident within a year, and (2) If an individual has had an accident, find the probability that they belong to the 'cautious' category.

Learn how to solve a probability problem involving risk classification in an insurance company. This example uses weighted averages and Bayes' Theorem to calculate the likelihood of an accident and conditional probabilities. Ideal for high school students in Grades 10-12.

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