An insurance agent offers his clients auto insurance, homeowners’ insurance and renters’ insurance. The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive. The profile of the agent’s clients is as follows:
a.	17% of the clients have none of these three products
b.	64% of the clients have auto insurance
c.	Twice as many of the clients have homeowners’ insurance as have renters’ insurance
d.	35% of the clients have two of these three products
e.	11% of the clients have homeowners’ insurance, but not auto insurance.
Calculate the percentage of the agent’s clients that have both auto and renters’ insurance.

Let's denote the following events:
- \( A \): clients with auto insurance
- \( H \): clients with homeowners’ insurance
- \( R \): clients with renters’ insurance

We are given:
1. \( P(\text{None}) = 0.17 \), meaning 17% of clients have none of the three products.
2. \( P(A) = 0.64 \), meaning 64% of clients have auto insurance.
3. \( P(H) = 2P(R) \), meaning twice as many clients have homeowners’ insurance as have renters’ insurance.
4. \( P(\text{exactly two products}) = 0.35 \), meaning 35% of clients have exactly two of the three products.
5. \( P(H \text{ and not } A) = 0.11 \), meaning 11% of clients have homeowners’ insurance but not auto insurance.

We need to calculate the percentage of the agent’s clients that have both auto and renters’ insurance, \( P(A \cap R) \).

### Step 1: Calculate \( P(H) \) and \( P(R) \)
Since \( P(H) = 2P(R) \), let \( P(R) = x \). Then, \( P(H) = 2x \).

### Step 2: Use the inclusion-exclusion principle
The total probability of clients having at least one of the three products is \( 1 - P(\text{None}) = 1 - 0.17 = 0.83 \).

The formula for the probability of at least one of the three products is:
\[
P(A \cup H \cup R) = P(A) + P(H) + P(R) - P(A \cap H) - P(A \cap R) - P(H \cap R) + P(A \cap H \cap R)
\]
Substituting the values we know:
\[
0.83 = 0.64 + 2x + x - P(A \cap H) - P(A \cap R) - P(H \cap R) + P(A \cap H \cap R)
\]
This simplifies to:
\[
0.83 = 0.64 + 3x - P(A \cap H) - P(A \cap R) - P(H \cap R) + P(A \cap H \cap R)
\]

### Step 3: Use the given \( P(H \text{ and not } A) = 0.11 \)
We know that \( P(H \text{ and not } A) = P(H) - P(A \cap H) = 0.11 \), and since \( P(H) = 2x \), we get:
\[
P(A \cap H) = 2x - 0.11
\]

### Step 4: Use the information about clients with exactly two products
We are given that 35% of clients have exactly two products:
\[
P(\text{exactly two}) = P(A \cap H) + P(A \cap R) + P(H \cap R) - 3P(A \cap H \cap R) = 0.35
\]
Substitute \( P(A \cap H) = 2x - 0.11 \) into this equation:
\[
0.35 = (2x - 0.11) + P(A \cap R) + P(H \cap R) - 3P(A \cap H \cap R)
\]

### Step 5: Solve for \( P(A \cap R) \)
We need additional information or assumptions to further solve this. However, we have an equation involving \( x \), \( P(A \cap R) \), and \( P(H \cap R) \).

Given that \( P(H) = 2x \) and \( P(R) = x \), let's assume values for \( x \) and solve, but to keep it general, let's express \( P(A \cap R) \) as a function of \( x \).

### Final Calculation:
By solving the above equations, \( P(A \cap R) \) can be derived. Given the complexity and typical values, let's assume reasonable values for \( x \) to solve, or proceed to numerical solving methods. 

For now, this remains as the derived system of equations that needs numeric or algebraic resolution.

If you have any clarifications or if you'd like to explore specific numeric solving methods, let me know!

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1. How would the calculations change if \( P(H) = P(R) \)?
2. What if we knew the value of \( P(A \cap H \cap R) \)?
3. Can we simplify the system if we know the distribution of \( P(A \cap H \cap R) \)?
4. What assumptions can we reasonably make for the given percentages?
5. How would the situation change if we added another insurance product?

**Tip:** Always break down multi-step problems into smaller parts and solve sequentially.

Learn how to solve a complex probability problem involving clients' insurance choices. Calculate the percentage of clients with both auto and renters' insurance using probability theory and the inclusion-exclusion principle.

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