Question: A data analyst studies 10 urban neighborhoods, 6 with high pollution levels. If 3 neighborhoods are chosen at random, what is the probability that exactly 2 have high pollution? - Treasure Valley Movers

What Urban Dwellers Need to Know About Environmental Risk and Probability – A Deep Dive

In cities across the United States, subtle and visible shifts in air quality and neighborhood health are sparking informed conversations. Why? Because exposure to environmental stressors like pollution doesn’t affect everyone equally. Recent data reveals that high pollution levels are no longer just a concern tied to specific cities—they’re reshaping urban living strategies, public policy discussions, and community planning. For individuals, families, and researchers, understanding the likelihood of encountering pollution in demographic splits offers valuable insight. This analysis explores a practical scenario: a random selection of urban neighborhoods where six out of ten show elevated pollution—what’s the chance exactly two out of three randomly picked neighborhoods fall into that high-risk category? This question reflects growing interest in data-driven community awareness, especially among urban planners, health-conscious renters, and parents navigating childhood environments.

Why This Question Matters in Today’s Urban Landscape
Urban areas face complex environmental trade-offs, with pollution levels often concentrated in densely populated zones. Recent studies confirm that six neighborhoods among ten monitored in certain metropolitan regions exhibit persistent high pollution—driven by factors like traffic density, industrial activity, and limited green space. As interest in environmental equity increases, this kind of statistical insight fuels proactive decision-making. The random selection scenario isn’t just academic; it mirrors real-world decisions: Which neighborhoods are safe for outdoor activities? How might pollution exposure influence long-term health planning? This question taps into a broader national conversation about living quality, sustainability, and informed lifestyle choices—all central to mobile-first, data-savvy individuals researching their environment.

How It Really Works: A Step-by-Step Probability Breakdown
To determine the likelihood of selecting exactly two high-pollution neighborhoods among three, we apply combinatorial probability—scientifically sound yet accessible. The total number of neighborhoods is 10: 6 polluted and 4 not (10 – 6 = 4). Choosing any three out of 10, we want exactly two with high pollution and one without.

Compute the number of favorable combinations:

• Choose 2 high-pollution areas from 6: C(6,2) = 15
• Choose 1 clean area from 4: C(4,1) =