#### 168.75**Question:** A seismologist is analyzing data from 7 different seismic stations. Each station can independently detect an earthquake with a probability of \( \frac{1}{2} \). What is the probability that exactly 4 out of the 7 stations detect an earthquake?

["Title: Probability That Exactly 4 Out of 7 Seismic Stations Detect an Earthquake | Seismology Analysis", "---", "When monitoring earthquake activity, seismologists rely on networks of monitoring stations to capture early signals. Understanding detection probabilities across multiple stations helps assess early warning reliability. In a key scenario, imagine 7 independent seismic stations, each detecting an earthquake with a fixed probability of ( \frac{1}{2} ). What is the probability that exactly 4 out of these 7 stations detect the earthquake?", "This article explores the probabilistic modeling behind this situation using the binomial distribution.", "---", "### Understanding the Problem with Binomial Probability", "Each seismic station operates as a Bernoulli trial: it either detects the earthquake (success) with probability ( p = \frac{1}{2} ), or fails to detect it (failure) with probability ( 1 - p = \frac{1}{2} ).", "Since the stations operate independently and each has the same success probability, the total number of stations detecting an earthquake follows a binomial distribution with parameters ( n = 7 ) (number of trials) and ( p = \frac{1}{2} ) (success probability per trial).", "We seek the probability of exactly ( k = 4 ) successes:", "[\nP(X = 4) = \binom{7}{4} \left( \frac{1}{2} \right)^4 \left( 1 - \frac{1}{2} \right)^{7-4}\n]", "---", "### Breaking Down the Formula", "- ( \binom{7}{4} ): This is the number of combinations to choose 4 detecting stations out of 7.\n [\n \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \ imes 6 \ imes 5}{3 \ imes 2 \ imes 1} = 35\n ]", "- ( \left( \frac{1}{2} \right)^4 = \frac{1}{16} ) — probability of 4 detections\n- ( \left( \frac{1}{2} \right)^3 = \frac{1}{8} ) — probability of 3 undetected stations", "Multiply all components:", "[\nP(X = 4) = 35 \ imes \frac{1}{16} \ imes \frac{1}{8} = 35 \ imes \frac{1}{128} = \frac{35}{128}\n]", "---", "### Final Result", "The probability that exactly 4 out of 7 independent seismic stations detect an earthquake—each with a 50% detection chance—is:", "[\n\boxed{\frac{35}{128} \approx 0.2734}\n]", "---", "### Why This Matters in Seismology", "Accurate probability modeling helps seismologists evaluate station reliability, optimize network layouts, and improve earthquake early warning systems. When each of 7 stations independently has a 50% detection likelihood, understanding the chance of exactly 4 detections ensures effective risk assessment and response planning.", "---", "Keywords: seismology, earthquake detection, binomial distribution, probability calculation, seismographic data analysis, 7 seismic stations, detection probability, early warning systems", "Meta Description: Explore the probability of exactly 4 out of 7 independent seismic stations detecting an earthquake using binomial statistics — essential for earthquake monitoring and early warning reliability."]