It is estimated that 88% of senior citizens suffer from steep disorders and 7% suffer from anxiety. Moreover, 5% of senior citizens suffer from both sleep disorders and anxiety. Find the probability that a senior citizen suffers from anxiety, given that he or she has sleep disorder. Round your answer to decimal places.

We are given the following information:

- \( P(S) = 0.88 \) (Probability that a senior citizen suffers from sleep disorders)
- \( P(A) = 0.07 \) (Probability that a senior citizen suffers from anxiety)
- \( P(S \cap A) = 0.05 \) (Probability that a senior citizen suffers from both sleep disorders and anxiety)

We need to find the conditional probability that a senior citizen suffers from anxiety, given that he or she has a sleep disorder. This is denoted by \( P(A|S) \).

The formula for conditional probability is:

\[
P(A|S) = \frac{P(S \cap A)}{P(S)}
\]

Substituting the given values:

\[
P(A|S) = \frac{0.05}{0.88}
\]

Now, let's calculate the value:

\[
P(A|S) \approx 0.0568
\]

So, the probability that a senior citizen suffers from anxiety, given that he or she has a sleep disorder, is approximately **0.057** (rounded to three decimal places).

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

Here are 8 related questions you might find useful:

1. How do you interpret the result of a conditional probability calculation?
2. What are some common misconceptions about conditional probability?
3. Can conditional probability be greater than 1?
4. How does Bayes' theorem relate to conditional probability?
5. What if the percentages for anxiety and sleep disorders were different—how would the result change?
6. What real-life situations involve conditional probability?
7. What is the probability that a senior citizen suffers from either sleep disorders or anxiety (or both)?
8. How would the calculation change if we were dealing with multiple overlapping conditions?

**Tip:** Conditional probability is often used in medical diagnostics to assess the likelihood of a condition given the presence of certain symptoms.

Learn how to calculate the conditional probability that a senior citizen suffers from anxiety given they have a sleep disorder. Given probabilities include P(S) = 0.88, P(A) = 0.07, and P(S ∩ A) = 0.05. Understand the method using the conditional probability formula. Suitable for advanced high school students and anyone interested in probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation