Mathematics

Mathematics

Problem

For a certain drug, the probability that a single patient experiences relief is 0.050.05 and patients act independently. Determine the minimum number of patients that must be treated so that the probability of observing at least one relief case is at least 0.900.90.
Intuitive explanation

This is a binomial probability problem where you're hunting for a threshold. The key insight is to flip the question: instead of calculating "at least one success" directly, use the complement—find when "zero successes" becomes rare enough. Since each patient has a tiny success probability, you need many trials; set up the inequality 1−(0.95)n≥0.901 - (0.95)^n \geq 0.90 and solve for nn using logarithms.

Intuitive explanation

This is a binomial probability problem where you're hunting for a threshold. The key insight is to flip the question: instead of calculating "at least one success" directly, use the complement—find when "zero successes" becomes rare enough. Since each patient has a tiny success probability, you need many trials; set up the inequality 1−(0.95)n≥0.901 - (0.95)^n \geq 0.90 and solve for nn using logarithms.

Browse the problem library