A pharmaceutical research company is testing the effectiveness of a new antiviral treatment, therapy TTT. In a clinical trial involving a large cohort of patients, two events were recorded:
Event T T\,T is defined as the patient receiving the antiviral therapy.
Event R R\,R is defined as the patient showing a significant reduction in viral load within 24 hours.
The following table summarises the joint probabilities for these events based on the trial data:
| RRR | R′R'R′ | |
|---|---|---|
| TTT | 0.36 | 0.24 |
| T′T'T′ | 0.12 | 0.28 |
A patient from the trial is selected at random.
(i) Find P(T)P(T)P(T).
(ii) Find P((T∩R)′)P((T \cap R)')P((T∩R)′).
(iii) Find P(R∣T′)P(R | T')P(R∣T′).
Determine whether the events T T\,T and R R\,R are independent. Fully justify your answer.