Questions: It is estimated that 90% of senior citizens suffer from sleep disorders and 8% suffer from anxiety. Moreover, 6% of senior citizens suffer from both sleep disorders and anxiety. Answer the questions below. (If necessary, consult a list of formulas.) (a) Given that a senior citizen suffers from anxiety, what is the probability that he or she also suffers from sleep disorder? Round your answer to 2 decimal places. (b) Find the probability that a senior citizen suffers from anxiety, given that he or she has sleep disorder. Round your answer to 2 decimal places.

Solution Steps

Let \( A \) be the event that a senior citizen suffers from anxiety, and \( S \) be the event that a senior citizen suffers from sleep disorders. We are given the following probabilities:

• \( P(S) = 0.90 \)
• \( P(A) = 0.08 \)
• \( P(S \cap A) = 0.06 \)

We need to find the probability that a senior citizen suffers from sleep disorders given that they suffer from anxiety, \( P(S \mid A) \).

Using the formula for conditional probability: \[ P(S \mid A) = \frac{P(S \cap A)}{P(A)} \]

Substitute the given values: \[ P(S \mid A) = \frac{0.06}{0.08} = 0.75 \]

We need to find the probability that a senior citizen suffers from anxiety given that they have sleep disorders, \( P(A \mid S) \).

Using the formula for conditional probability: \[ P(A \mid S) = \frac{P(S \cap A)}{P(S)} \]

Substitute the given values: \[ P(A \mid S) = \frac{0.06}{0.90} \approx 0.0667 \]

Final Answer