Question: A geotechnical engineer is testing soil samples from 10 different sites, each labeled from 1 to 10. If 3 sites are selected at random for detailed analysis, what is the probability that site 7 is included?

["Understanding Probability in Geotechnical Testing: Calculating the Chance Site 7 Is Selected", "When conducting geotechnical investigations, precise sampling strategy is essential for accurate site assessment. A common question in probability-based sampling involves selecting a subset of sites from a larger set—particularly, what is the probability that a specific site, such as Site 7, is included when a random sample of 3 out of 10 sites is chosen for detailed soil analysis?", "### The Scenario: Choosing 3 Sites from 10", "Imagine a scenario where a geotechnical engineer collects soil samples from 10 distinct locations, labeled Site 1 through Site 10. The sampling process randomly selects exactly 3 sites for in-depth testing. To assess the likelihood that Site 7 is among these selected sites, we apply fundamental combinatorial probability principles.", "### Why This Matters", "Understanding the probability that a specific site—such as Site 7—is included helps engineers optimize sampling efficiency, plan fieldwork logistics, and ensure representative data coverage. It also enhances decision-making for risk assessment and site development.", "### How to Calculate the Probability", "To determine the probability that Site 7 is selected:", "1. Total Possible Selections:\n The total number of ways to choose 3 sites from 10 is given by the combination formula:\n $$ \binom{10}{3} = \frac{10!}{3!(10-3)!} = 120 $$", "2. Favorable Outcomes (Site 7 Included):\n If Site 7 is selected, we only need to choose the remaining 2 sites from the other 9 (excluding Site 7). The number of ways to do this is:\n $$ \binom{9}{2} = \frac{9!}{2!(9-2)!} = 36 $$", "3. Probability Calculation:\n The probability that Site 7 is included is the ratio of favorable outcomes to total possible outcomes:\n $$ P(\ ext{Site 7 included}) = \frac{\binom{9}{2}}{\binom{10}{3}} = \frac{36}{120} = \frac{3}{10} = 0.3 \ ext{ or } 30% $$", "### Conclusion", "There is a 30% probability that Site 7 is selected when 3 sites are randomly chosen from 10 for geotechnical testing. This probabilistic insight supports systematic sampling design, ensuring both statistical validity and efficient resource use in engineering projects.", "By leveraging combinatorial methods, geotechnical engineers enhance the rigor of their site assessments—proving that even in soil analysis, probability plays a crucial role in precision and probability.", "Keywords: geotechnical engineer, soil sampling probability, Site 7 inclusion, combinatorics, probability calculation, geotechnical testing, sampling strategy, root cause analysis, engineering probability."]