Question: A box contains 7 red sensors and 9 blue sensors. If a technician randomly selects 5 sensors without replacement, what is the probability that exactly 3 are red and 2 are blue?

["Understanding the Probability of Selecting Sensors: A Comprehensive Guide", "When tasked with a probability question like “A box contains 7 red sensors and 9 blue sensors. If a technician randomly selects 5 sensors without replacement, what is the probability that exactly 3 are red and 2 are blue?”, understanding the underlying concepts can transform confusion into clarity. In this article, we’ll break down the problem step-by-step and explain the key probability principles involved, including combinations, hypergeometric distribution, and conditional reasoning—helping you tackle similar problems with confidence.", "---", "### The Probability Problem At a Glance", "We have a box with 7 red sensors and 9 blue sensors, totaling 16 sensors. A technician draws 5 sensors at random without replacement. We want the probability that exactly 3 are red and 2 are blue.", "---", "### Why Not a Simple Frequence Dip?", "One might mistakenly think in terms of simple fractions, but because the sensors are drawn without replacement, the probability changes with each selection. This scenario follows the hypergeometric distribution, which models success rates in sampling without replacement from a finite population.", "---", "### Key Concepts Explained", "#### 1. Hypergeometric Distribution Formula", "The probability of drawing exactly k successes (in this case, red sensors) in n draws without replacement from a population of size N containing K successes is:", "[\nP(X = k) = \frac{{\binom{K}{k} \ imes \binom{N-K}{n-k}}}{\binom{N}{n}}\n]", "Where:\n- ( N = 16 ): total sensors\n- ( K = 7 ): total red sensors (successes)\n- ( n = 5 ): number of draws\n- ( k = 3 ): desired red sensors", "#### 2. Interpreting the Formula", "- Numerator:\n - (\binom{7}{3}): ways to choose 3 red sensors from 7\n - (\binom{9}{2}): ways to choose 2 blue sensors from 9\n - Multiplying them gives favorable outcomes (3 red, 2 blue)", "- Denominator:\n - (\binom{16}{5}): total possible ways to choose any 5 sensors from 16", "---", "### Step-by-Step Calculation", "Let’s compute each combination:", "1. Ways to choose 3 red sensors from 7:\n[\n\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \ imes 6 \ imes 5}{3 \ imes 2 \ imes 1} = 35\n]", "2. Ways to choose 2 blue sensors from 9:\n[\n\binom{9}{2} = \frac{9 \ imes 8}{2 \ imes 1} = 36\n]", "3. Multiply favorable combinations:\n[\n35 \ imes 36 = 1260\n]", "4. Total ways to choose any 5 sensors from 16:\n[\n\binom{16}{5} = \frac{16 \ imes 15 \ imes 14 \ imes 13 \ imes 12}{5 \ imes 4 \ imes 3 \ imes 2 \ imes 1} = 4368\n]", "5. Final probability:\n[\nP(3 \ ext{ red, } 2 \ ext{ blue}) = \frac{1260}{4368}\n]", "6. Simplify the fraction:\nDivide numerator and denominator by 12:\n[\n\frac{1260 \div 12}{4368 \div 12} = \frac{105}{364}\n]", "This fraction can be simplified further if needed, but (\frac{105}{364}) is an accepted reduced form for practical purposes. As a decimal:\n[\n\frac{105}{364} \approx 0.2887 \ ext{ or } 28.87%\n]", "---", "### Why Correct Probability Matters", "Understanding this probability isn’t just academic. In real-world scenarios—such as quality assurance in electronics manufacturing, sensor deployment in IoT systems, or reliability testing—technicians rely on accurate probability models to assess risk, plan maintenance, and optimize workflows.", "---", "### Final Thoughts", "To master probability questions like this:", "- Identify whether sampling is with or without replacement.\n- Recognize the hypergeometric distribution model.\n- Compute favorable outcomes over total possible outcomes using combinations.\n- Always simplify and interpret results in context.", "With practice, these structured steps demystify even complex probability problems—empowering you to analyze data-driven scenarios with confidence and precision.", "---", "Want to master probability? Try another similar problem—like drawing colored chips, selecting balls from a bag, or drawing cards—and apply the same formula logic. The more you practice, the sharper your intuition becomes!"]