Question

A virus has two variants P and Q. Two treatments L and M were developed against variant P and Q and tested through four independent trials. The table shows the results. Assuming that both variants are equally prevalent, and both treatments cost about the same, but a given patient can be treated with only one treatment, which of the options is/are true?

• A. If the patient is known to be infected by variant P, both the treatments will be equally effective
• B. If we can determine that the patient is infected by variant Q, treatment L has a higher chance of success
• C. If the variant that the patient is infected by is not known, then both the treatments can be equally effective
• D. If a hospital can stock only one of the two treatments, they should stock treatment L

Hint:

In data interpretation questions, always pay close attention to the assumptions given in the problem statement (e.g., "equally prevalent"). These assumptions are crucial and can override information that seems apparent from summary columns in a table.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question requires us to calculate and compare the success rates (probabilities of success) of two different treatments against two different virus variants. We need to evaluate each statement based on these calculated probabilities. A key piece of information is the assumption that both variants are ""equally prevalent,"" meaning the probability of a random patient having variant P is 0.5, and variant Q is 0.5. 

Step 2: Key Formula or Approach: 
The success rate for each trial is calculated as: \[ \text{Success Rate} = \frac{\text{Number of Successes}}{\text{Total Number of Trials}} \] For cases where the variant is unknown, the overall effectiveness of a treatment is the weighted average of its success rates against each variant, with the weights being the prevalence of each variant. \[ \text{Overall Effectiveness} = (\text{Success Rate for P} \times \text{Prevalence of P}) + (\text{Success Rate for Q} \times \text{Prevalence of Q}) \] 
Step 3: Detailed Explanation: 
First, let's calculate the success rate for each treatment against each variant: 
Success Rate of L against P: \( \frac{44}{50} = 0.88 \) or 88%. 
Success Rate of L against Q: \( \frac{81}{100} = 0.81 \) or 81%. 
Success Rate of M against P: \( \frac{94}{100} = 0.94 \) or 94%. 
Success Rate of M against Q: \( \frac{31}{50} = 0.62 \) or 62%. 
Now, let's evaluate each option: 
(A) If the patient is known to be infected by variant P, both the treatments will be equally effective. 
Comparing the success rates for variant P: Treatment L has an 88% success rate. 
Treatment M has a 94% success rate. 
Since 94% $>$ 88%, Treatment M is more effective against variant P. Thus, statement (A) is false. 
(B) If we can determine that the patient is infected by variant Q, treatment L has a higher chance of success. 
Comparing the success rates for variant Q: Treatment L has an 81% success rate. 
Treatment M has a 62% success rate. 
Since 81% $>$ 62%, Treatment L does have a higher chance of success against variant Q. Thus, statement (B) is true. 
(C) If the variant that the patient is infected by is not known, then both the treatments can be equally effective. 
We use the overall effectiveness formula with the assumption that P(P) = P(Q) = 0.5. 
Overall Effectiveness of L = \( (0.88 \times 0.5) + (0.81 \times 0.5) = 0.44 + 0.405 = 0.845 \) or 84.5%. 
Overall Effectiveness of M = \( (0.94 \times 0.5) + (0.62 \times 0.5) = 0.47 + 0.31 = 0.78 \) or 78%. 
Since 84.5% \( \neq \) 78%, the treatments are not equally effective when the variant is unknown. Treatment L is more effective overall. The ""Total"" column in the table is based on the number of patients in the trials (150 for each), not on an equally prevalent population, which is why it is misleading. Thus, statement (C) is false. 
(D) If a hospital can stock only one of the two treatments, they should stock treatment L. 
To make this decision, the hospital should choose the treatment with the higher overall effectiveness for a random patient (where the variant is unknown). Based on our calculation for option (C), Treatment L has an overall effectiveness of 84.5%, while Treatment M has an overall effectiveness of 78%. Since L is more effective overall, the hospital should stock Treatment L. Thus, statement (D) is true. 

Step 4: Final Answer: 
Based on the analysis, statements (B) and (D) are true.