Suppose that an insurance company classifies people who buy medical insurance from them
into one of three classes: good, average, and bad risks. As a data scientist for the company,
you have access to the following customer data for the calendar year 2023-24:
Class % of customers % who had a major health problem
Bad risk 20% 25%
Average risk 40% 20%
Good risk 40% 10%
What is the probability that a new customer will not meet with a major health problem
during 2024-25?

To find the probability that a new customer will **not** meet with a major health problem during the year 2024-25, we can use the total probability rule, which takes into account the different risk categories (good, average, and bad) and their associated probabilities.

### Step-by-step approach:

1. **Define events:**
   - Let \( A \) be the event that a customer does **not** have a major health problem.
   - Let \( G \), \( A_v \), and \( B \) represent the events that a customer is classified as a **good risk**, **average risk**, or **bad risk**, respectively.

2. **Information given:**
   - Probability of each class (prior probabilities):
     - \( P(G) = 0.40 \) (Good risk)
     - \( P(A_v) = 0.40 \) (Average risk)
     - \( P(B) = 0.20 \) (Bad risk)
   - Conditional probabilities of having a major health problem:
     - \( P(\text{Major health problem} | G) = 0.10 \)
     - \( P(\text{Major health problem} | A_v) = 0.20 \)
     - \( P(\text{Major health problem} | B) = 0.25 \)

3. **Complementary probabilities (no major health problem):**
   - \( P(A | G) = 1 - 0.10 = 0.90 \)
   - \( P(A | A_v) = 1 - 0.20 = 0.80 \)
   - \( P(A | B) = 1 - 0.25 = 0.75 \)

4. **Use the law of total probability**:
   The total probability that a new customer does not have a major health problem is given by:
   \[
   P(A) = P(A | G) \cdot P(G) + P(A | A_v) \cdot P(A_v) + P(A | B) \cdot P(B)
   \]
   Substituting the values:
   \[
   P(A) = (0.90 \times 0.40) + (0.80 \times 0.40) + (0.75 \times 0.20)
   \]
   \[
   P(A) = 0.36 + 0.32 + 0.15
   \]
   \[
   P(A) = 0.83
   \]

### Conclusion:
The probability that a new customer will not meet with a major health problem during 2024-25 is **0.83** or **83%**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the probability that a new customer **will** meet with a major health problem?
2. How would the result change if the proportions of customers in each class were different?
3. What is the significance of using the total probability rule in this context?
4. How could the insurance company use this information to set premium rates?
5. How can machine learning models predict a customer’s risk based on more granular data?

**Tip:** The total probability rule is often used in cases where outcomes depend on different underlying conditions, such as risk classes.

Suppose that an insurance company classifies people who buy medical insurance from them into one of three classes: good, average, and bad risks. As a data scientist for the company, you have access to the following customer data for the calendar year 2023-24: Class % of customers % who had a major health problem Bad risk 20% 25% Average risk 40% 20% Good risk 40% 10% What is the probability that a new customer will not meet with a major health problem during 2024-25?

Discover how to calculate the probability that a new customer will not face a major health problem during 2024-25. This problem applies probability theory, specifically the law of total probability and conditional probability, to a real-world insurance classification scenario. Understand the risk percentages and their impact on health outcomes.

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