Question: A historian studies three scientific breakthroughs from a collection of 20, where 5 are pivotal. What is the probability that exactly two of the selected breakthroughs are pivotal?
["Title: Understanding the Probability: Selecting Exactly Two Pivotal Scientific Breakthroughs from 20", "Meta Description: Discover the probability of selecting exactly two pivotal scientific breakthroughs when studying three out of 20, where 5 are pivotal. Learn how combinatorics powers this analysis.", "---", "Question: A historian studies three scientific breakthroughs chosen from a collection of 20, of which 5 are pivotal. What is the probability that exactly two of the selected breakthroughs are pivotal?", "---", "When delving into history’s milestones, historians often face the challenge of analyzing selection from a large set of achievements—especially when distinguishing pivotal innovations from notable but less transformative ones. A common statistical question arises: What is the probability that among three randomly selected scientific breakthroughs from 20, exactly two are pivotal? This problem blends historical inquiry with probability and combinatorics, offering a clear framework for applying conditional chance and binomial logic.", "### The Core Problem Breakdown", "You’re selecting 3 breakthroughs from a total pool of 20, of which 5 are pivotal. Among these, the key is to compute the probability that exactly 2 of the 3 selected breakthroughs are pivotal.", "This is a classic hypergeometric probability scenario—sampling without replacement from a finite population with two distinct categories: pivotal vs. non-pivotal. Here’s the detailed breakdown:", "---", "### Step 1: Define the Groups\n- Total scientific breakthroughs: 20\n- Pivotal breakthroughs: 5\n- Non-pivotal breakthroughs: 15", "---", "### Step 2: Use the Hypergeometric Probability Formula", "The probability of selecting exactly ( k ) pivotal breakthroughs in a sample of size ( n ) from a total with ( K ) pivotal items follows:", "[\nP(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}\n]", "Where:\n- ( N = 20 ) (total breakthroughs)\n- ( K = 5 ) (pivotal ones)\n- ( n = 3 ) (selected breakthroughs)\n- ( k = 2 ) (exactly two pivotal)", "Plugging in the values:", "[\nP(X = 2) = \frac{\binom{5}{2} \binom{15}{1}}{\binom{20}{3}}\n]", "---", "### Step 3: Compute Each Component", "- (\binom{5}{2} = \frac{5!}{2!(5-2)!} = 10)\n- (\binom{15}{1} = 15)\n- (\binom{20}{3} = \frac{20 \ imes 19 \ imes 18}{3 \ imes 2 \ imes 1} = 1140)", "Now calculate:", "[\nP(X = 2) = \frac{10 \ imes 15}{1140} = \frac{150}{1140} = \frac{15}{114} = \frac{5}{38}\n]", "---", "### Final Result", "The probability that exactly two of the three selected breakthroughs are pivotal is (\frac{5}{38} \approx 0.1316), or about 13.16%.", "---", "### Why This Matters for Historians", "Understanding such probabilities helps historians better appreciate the composition and significance of scientific collections. By quantifying how likely it is that at least two of three sampled achievements are truly transformative (pivotal), researchers can design more nuanced historical analyses, assess bias in historical documentation, and contextualize breakthroughs more accurately within their broader impact.", "---", "### Summary", "- Sampling 3 from 20 with 5 pivotal\n- Use hypergeometric probability\n- Result: exactly 2 pivotal breakthroughs has a probability of (\frac{5}{38})\n- This approach supports rigorous, data-driven historical inquiry", "---", "Keywords: probability, historian, scientific breakthroughs, hypergeometric distribution, combinatorics, pivotal discoveries, statistical analysis, historical methodology, selection probability, combination formulas.", "---", "Explore how probability transforms historical research and enables deeper insights into the evolution of science."]
