It is estimated that $92 \%$ of senior citizens suffer from sleep disorders and $7 \%$ suffer from anxiety. Moreover, $5 \%$ 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. ㅁ
Solution
Step 1 ：We are given that the probability of a senior citizen suffering from sleep disorder, denoted as P(A), is 0.92.
Step 2 ：The probability of a senior citizen suffering from anxiety, denoted as P(B), is 0.07.
Step 3 ：The probability of a senior citizen suffering from both sleep disorder and anxiety, denoted as P(A ∩ B), is 0.05.
Step 4 ：We can calculate the probability that a senior citizen suffers from sleep disorder given that he or she suffers from anxiety using the formula for conditional probability: P(A|B) = P(A ∩ B) / P(B). Substituting the given values, we get P(A|B) = 0.05 / 0.07 = 0.7142857142857143.
Step 5 ：Rounding to two decimal places, we get P(A|B) = \(\boxed{0.71}\).
Step 6 ：We can also calculate the probability that a senior citizen suffers from anxiety given that he or she suffers from sleep disorder using the formula for conditional probability: P(B|A) = P(A ∩ B) / P(A). Substituting the given values, we get P(B|A) = 0.05 / 0.92 = 0.05434782608695652.
Step 7 ：Rounding to two decimal places, we get P(B|A) = \(\boxed{0.05}\).
