A pharmacologist is testing a new drug and finds that it has a 70% chance of being effective on a single patient. If the drug is administered to four independent patients, what is the probability that the drug is effective on exactly two of them? - Treasure Valley Movers
A Pharmacologist’s Data: What’s the Chance a New Drug Works on Exactly Two Out of Four Patients?
A Pharmacologist’s Data: What’s the Chance a New Drug Works on Exactly Two Out of Four Patients?
When breakthrough treatments enter clinical trials, the numbers behind success can spark quiet curiosity. A pharmacologist recently reported a 70% effectiveness rate for a new drug tested on four independent patients. This figure alone raises compelling questions: What’s the likelihood the drug shows benefit in exactly two individuals? And why does this simple statistic matter to researchers, patients, and the broader healthcare landscape?
This query isn’t just technical—it reflects a growing public and professional interest in meaningful treatment outcomes. As medical advancements accelerate, people are more aware of precision medicine and patient-specific responses. A drug with a 70% success rate invites deeper inquiry into probability, risk, and real-world application. For those navigating health decisions or researching innovations, understanding these probabilities supports informed expectations.
Understanding the Context
The Math Behind the Probability
To calculate the chance the drug works on exactly two of four patients, researchers use the binomial probability formula. This model applies when outcomes are independent and fall into two categories—“effective” or “not effective”—each with consistent odds.
In this case, the effectiveness probability is 0.7, so failure occurs with probability 0.3. The trial involves four patients, and we seek the exact scenario where two succeed and two fail. The binomial coefficient determines how many ways two successes can occur among four trials:
- Number of combinations: ⁴C₂ = 6
- Probability of two successes: 0.7² = 0.49
- Probability of two failures: 0.3² = 0.09
Combined: (⁴C₂) × (0.7²) × (0.3²) = 6 × 0.49 × 0.09 = 0.2646
That’s approximately 26.5%—meaning a 70% single-patient success rate leads to about a 26.5% chance the drug works effectively on exactly
