Solution: Use complementary probability. The probability no sensor triggers is $ \left(\frac{3}{4}\right)^3 = \frac{27}{64} $. Subtract from 1:

["Title: Understanding Complementary Probability: How Sensors Work Together to Improve Reliability", "Meta Description:\nDiscover how complementary probability enhances sensor reliability by calculating the chance at least one sensor triggers. Learn why if each sensor has a 25% failure rate, using complementary events simplifies the solution.", "---", "Introduction\nIn systems relying on multiple sensors—from industrial automation to medical monitoring—ensuring accurate detection is critical. But when every sensor has a risk of failing, how do we confidently determine the likelihood of a detection event? This is where complementary probability becomes a powerful tool.", "Instead of directly calculating the probability that at least one sensor detects an event, complementary probability lets us compute the opposite scenario: the probability that no sensor triggers—a simpler case to analyze. Once understood, subtracting this probability from 1 delivers a clear answer.", "---", "What Is Complementary Probability?\nComplementary probability refers to the strategy of finding the probability that an event does not occur, by calculating the probability that the event does occur and subtracting it from 1.", "For independent events—events that don’t influence each other—this approach can drastically simplify complex calculations. In sensor networks, this means assessing the “non-failure” case and flipping it to find the real-world likelihood of detection.", "---", "A Real-World Example: Sensor Reliability\nSuppose a monitoring system uses three identical sensors, each with a 25% chance of failing (i.e., not triggering when an event occurs). This means each sensor has a 75% probability of working, or $ P(\ ext{sensor triggers}) = \frac{3}{4} $.", "Instead of computing the probability that “at least one sensor triggers,” we apply complementary probability using the simpler case:", "- Let $ A $ be the event “at least one sensor triggers.”\n- Then, $ A^c $ (the complement) is “no sensor triggers.”", "Since sensors act independently,\n$$\nP(\ ext{no sensor triggers}) = \left(\frac{1}{4}\right)^3 = \frac{1}{64}\n$$\nBut wait—this assumes each sensor fails with probability $ \frac{1}{4} $, not $ \frac{3}{4} $. Let’s double-check:", "If $ P(\ ext{failure}) = \frac{1}{4} $, then $ P(\ ext{working}) = 1 - \frac{1}{4} = \frac{3}{4} $.\nThus,\n$$\nP(\ ext{no sensor triggers}) = \left(\frac{1}{4}\right)^3 = \frac{1}{64}\n$$", "Now subtract from 1:\n$$\nP(\ ext{at least one sensor triggers}) = 1 - \frac{1}{64} = \frac{63}{64}\n$$", "---", "Why This Method Works So Well\nUsing complementary probability avoids cumbersome sums over multiple cases—like calculating $ P(1 \ ext{ trigger}) + P(2 \ ext{ triggers}) + P(3 \ ext{ triggers}) $—which requires binomial expansions. Instead, we focus solely on the unlikely, yet deterministic, case of total failure.", "In high-stakes applications such as aerospace, medical devices, or autonomous vehicles, this method ensures faster, more reliable risk assessment—critical for fail-safe design.", "---", "Conclusion\nComplementary probability transforms complex detection reliability problems into straightforward computations. By evaluating the risk of sensor failure together and subtracting from 1, engineers and data scientists gain a powerful, intuitive tool for building safer, smarter systems.", "So next time you analyze sensor networks or validate system reliability, remember: sometimes the easiest path to truth lies in calculating what’s not happening.", "---", "Keywords: complementary probability, sensor reliability, probability calculation, no sensor triggers, probability complementary events, fault-tolerant systems, risk assessment, failure probability, industrial sensors, system safety, probability theory", "---", "Further Reading:\n- How to calculate independent events in probability\n- Applying complementary probability in risk analysis\n- Statistical methods for sensor fault detection", "---\nWhy trust complementary probability? When lives and system integrity depend on accurate risk modeling—simplicity meets precision."]