is estimated that 28% of all adults in the United States invest in stocks and that 86% of U.S. adults have investments in fixed income instruments (savings accounts, bonds, etc.). It is also estimated that 25% of U.S. adults have investments in both stocks and fixed income instruments. What is the probability that a randomly chosen U.S. adult Invests in stocks, given that s/he invests in fixed Income instruments? Round your answer to 2 decimal places.

To solve this problem, we need to calculate the conditional probability that a randomly chosen U.S. adult invests in stocks, given that they invest in fixed income instruments.

Let's define the events as follows:
- \( A \): The event that a U.S. adult invests in stocks.
- \( B \): The event that a U.S. adult invests in fixed income instruments.

We are given the following probabilities:
- \( P(A) = 0.28 \) (the probability that a U.S. adult invests in stocks),
- \( P(B) = 0.86 \) (the probability that a U.S. adult invests in fixed income instruments),
- \( P(A \cap B) = 0.25 \) (the probability that a U.S. adult invests in both stocks and fixed income instruments).

The conditional probability that a U.S. adult invests in stocks given that they invest in fixed income instruments is denoted as \( P(A \mid B) \) and is calculated using the formula:

\[
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
\]

Substituting the given values:

\[
P(A \mid B) = \frac{0.25}{0.86} \approx 0.2907
\]

Rounding to two decimal places, the probability is approximately \( 0.29 \).

So, the probability that a randomly chosen U.S. adult invests in stocks given that they invest in fixed income instruments is **0.29**.

Would you like more details or have any questions?

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**Related Questions:**
1. What is the probability that a randomly chosen U.S. adult invests in both stocks and fixed income instruments?
2. If a U.S. adult does not invest in fixed income instruments, what is the probability that they invest in stocks?
3. How would the probability change if 90% of U.S. adults had investments in fixed income instruments?
4. What is the probability that a U.S. adult does not invest in either stocks or fixed income instruments?
5. How can the conditional probability be interpreted in a real-world scenario?
6. What is the joint probability of an adult investing in neither stocks nor fixed income instruments?
7. How can Venn diagrams help in visualizing the probabilities in this problem?
8. How does the independence of events affect the conditional probability?

**Tip:** Remember that conditional probability helps refine our predictions based on known information. It’s useful in various fields, from finance to medicine.

Learn how to calculate the probability that a randomly chosen U.S. adult invests in stocks given they invest in fixed income instruments. Detailed solution with probabilities provided. Suitable for college-level students and professionals interested in probability and finance.

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