Question: A town is evaluating air quality sensors placed in 7 neighborhoods. If 3 sensors are randomly activated for calibration today, what is the probability that at least one sensor from the high-risk zone (which has 2 sensors) is selected? - Treasure Valley Movers
A town is evaluating air quality sensors placed in 7 neighborhoods. If 3 sensors are randomly activated for calibration today, what is the probability that at least one sensor from the high-risk zone (which has 2 sensors) is selected?
A town is evaluating air quality sensors placed in 7 neighborhoods. If 3 sensors are randomly activated for calibration today, what is the probability that at least one sensor from the high-risk zone (which has 2 sensors) is selected?
In an era when urban air quality is a growing concern, communities across the United States are increasingly monitoring environmental health through smart infrastructure—such as networks of air quality sensors. A key question shaping local planning conversations is: when calibrating dozens of sensors across neighborhoods, what’s the chance that at least one from a high-risk zone—a designated area with elevated pollution levels—gets activated for routine checks? With two high-risk sensors among seven total, understanding the math behind this scenario reveals not only probability fundamentals but also insights into urban preparedness and risk distribution.
Why This Question Matters — Public Awareness & Smart Infrastructure
Understanding the Context
As cities invest in environmental monitoring to protect public health, the mechanics behind sensor networks have become more transparent. These devices collect real-time data vital for identifying pollution hotspots and guiding policy. The random activation of sensors for calibration ensures accuracy but also sparks interest—especially in neighborhoods marked as high-risk for air quality. The public increasingly expects detail behind these decisions, and questions like this reflect growing demand for data-driven accountability. When communities hear how probabilities become part of infrastructure planning, trust in local governance grows.
How the Probability Works: A Clear, Step-by-Step Look
To calculate the likelihood that at least one of the two high-risk sensors is selected when three are activated from seven total, it’s useful to examine the complementary scenario—calculating the chance that none of the two high-risk sensors are activated, then subtracting from one.
- Total sensors: 7
- High-risk sensors: 2
- Calibration sensors activated: 3
Key Insights
First, determine total ways to choose 3 sensors from 7:
C(7,3) = 35
Next, count combinations where none of the high-risk sensors are selected—only from the 5 remaining low-risk sensors:
C(5,3) = 10
Thus, the number of selections excluding both high-risk sensors is 10.
The probability of selecting none from the high-risk zone is:
10 / 35 = 2/7 ≈ 0.2857
Therefore, the probability that at least one high-risk sensor is selected becomes:
1 – (2/7) = 5/7 ≈ 0.7143 — or about 71.4%
This result shows a strong likelihood, reinforcing that sensors in high-risk zones carry meaningful representation in the calibration process.
