5Question: An AI startup founder is analyzing data from 12 sensors placed in agricultural fields, 4 of which are faulty. If the founder randomly selects 5 sensors for immediate diagnostics, what is the probability that exactly 2 of them are faulty?

Fri, 01 May 2026 02:24:52 +0000

**How Data Patterns Shape Smart Farming Decisions in 2025** As U.S. agriculture embraces AI-driven precision tools, understanding sensor data accuracy becomes critical for optimizing yields and reducing waste. A common challenge involves balancing limited testing resources with the need for reliable diagnostics—particularly when a small subset of devices may be compromised. This scenario, involving 12 field sensors with 4 known faulty units, mirrors real-world concerns about data integrity in critical tech deployments. Readers interested in agricultural innovation will find this probability problem a key lens for evaluating risk, planning diagnostics, and improving operational efficiency. **Why This Probability Matters in Modern Farm Management**
The agricultural sector increasingly depends on sensor networks to monitor soil moisture, temperature, and equipment integrity. When manual checks are impractical due to scale, automated sampling becomes essential—but faulty sensors skew results, risking costly misjudgments. This problem exemplifies a core issue: with 4 out of 12 sensors unreliable, selecting 5 for diagnostics introduces measurable uncertainty. Understanding the probability of exactly 2 faulty units allows founders to anticipate data quality, allocate resources wisely, and refine sensor deployment strategies grounded in statistical insight. **The Math Behind Data Reliability in Sensor Networks**
To determine the likelihood of diagnosing exactly 2 faulty sensors among 5 randomly selected, we apply combinatorics. With 4 faulty and 8 working sensors, the total sensor pool is 12. Choosing 5 total configurations gives (12 choose 5) possible groups. For exactly 2 faulty, we select 2 from the faulty group (4 available) and 3 from the working group (8 available). This combination—(4 choose 2) × (8 choose 3)—reveals the number of favorable outcomes.