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Let's solve this problem using the principle of inclusion-exclusion.

### Given Information
- Probability of investing in stocks, \( P(\text{Stocks}) = 0.56 \)
- Probability of investing in bonds, \( P(\text{Bonds}) = 0.33 \)
- Probability of investing in either stocks or bonds (or both), \( P(\text{Stocks or Bonds}) = 0.76 \)

### Objective
We need to find the probability that a person invests in both stocks and bonds, \( P(\text{Stocks and Bonds}) \).

### Using Inclusion-Exclusion Principle
The formula for the probability of the union of two events \( A \) and \( B \) is:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
In this case:
\[
P(\text{Stocks or Bonds}) = P(\text{Stocks}) + P(\text{Bonds}) - P(\text{Stocks and Bonds})
\]

Rearranging to solve for \( P(\text{Stocks and Bonds}) \):
\[
P(\text{Stocks and Bonds}) = P(\text{Stocks}) + P(\text{Bonds}) - P(\text{Stocks or Bonds})
\]

### Plugging in the Values
\[
P(\text{Stocks and Bonds}) = 0.56 + 0.33 - 0.76
\]
\[
P(\text{Stocks and Bonds}) = 0.89 - 0.76
\]
\[
P(\text{Stocks and Bonds}) = 0.13
\]

### Answer
The probability that a person will invest in both stocks and bonds is **0.13**.

Would you like further details on this calculation or have any questions?

Here are some related questions to explore:

1. How would the probability change if the percentage of those investing in stocks increased?
2. Can you use a Venn diagram to illustrate this type of problem?
3. How does the principle of inclusion-exclusion work for three or more events?
4. What is the probability that a person will invest in only stocks?
5. What is the probability that a person will invest in only bonds?

**Tip:** The inclusion-exclusion principle is essential for calculating the union of overlapping events, especially in probability and set theory.

Suppose that 56% of the people who inquire about investments at a certain brokerage firm end up investing in stocks, 33% end up investing in bonds, and 76% end up investing in stocks or bonds (or both). What is the probability that a person who inquires about investments at this firm will invest in both stocks and bonds? Write your answer as a decimal (not as a percentage).

Learn how to calculate the probability that a person who inquires about investments will invest in both stocks and bonds. This problem uses the inclusion-exclusion principle to find overlapping probabilities, given that 56% invest in stocks, 33% in bonds, and 76% in stocks or bonds. Suitable for students in Grades 11-12.

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