Question

question
doctors conducting a pharmaceutical study release the following information:

• the probability that a patient will receive a trial medication in the first round is \\(\frac{11}{40}\\);
• the probability that a patient will receive a placebo medication in the second round given that they received a trial medication in the first round is \\(\frac{1}{3}\\);
• the probability that a patient will receive a placebo medication in the second round is \\(\frac{7}{10}\\).

let event \\(a\\) be the event that a patient receives a trial medication in the first round and event \\(b\\) be the event that a patient receives a placebo medication in the second round. are events \\(a\\) and \\(b\\) mutually exclusive, independent, both, or neither?
select the correct answer below:
\\(\bigcirc\\) events \\(a\\) and \\(b\\) are mutually exclusive.
\\(\bigcirc\\) events \\(a\\) and \\(b\\) are independent.
\\(\bigcirc\\) events \\(a\\) and \\(b\\) are neither mutually exclusive nor independent.
\\(\bigcirc\\) events \\(a\\) and \\(b\\) are both independent and mutually exclusive.

Explanation:

Step1: Recall definitions of mutually exclusive and independent events

• Mutually exclusive: \( P(A \cap B) = 0 \) (events cannot occur at the same time).
• Independent: \( P(A \cap B) = P(A) \times P(B) \) (occurrence of one does not affect the other).

Step2: Calculate \( P(A \cap B) \) using conditional probability

The formula for conditional probability is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \), so \( P(A \cap B) = P(B|A) \times P(A) \).
Given \( P(A) = \frac{11}{40} \) and \( P(B|A) = \frac{1}{3} \), we have:
\( P(A \cap B) = \frac{1}{3} \times \frac{11}{40} = \frac{11}{120} \)

Step3: Check if mutually exclusive

Since \( P(A \cap B) = \frac{11}{120}
eq 0 \), events \( A \) and \( B \) are not mutually exclusive.

Step4: Check if independent

Calculate \( P(A) \times P(B) \). Given \( P(B) = \frac{7}{10} \), we have:
\( P(A) \times P(B) = \frac{11}{40} \times \frac{7}{10} = \frac{77}{400} \)
Now, compare with \( P(A \cap B) = \frac{11}{120} \approx 0.0917 \) and \( \frac{77}{400} = 0.1925 \). Since \( \frac{11}{120}
eq \frac{77}{400} \), events \( A \) and \( B \) are not independent.

Answer:

Events \( A \) and \( B \) are neither mutually exclusive nor independent.