A factory produces widgets with a defect rate of 3%. A quality inspector selects a random sample of 150 widgets. What is the probability that exactly 5 are defective?

["Understanding the Probability of Defects: A Widget Factory’s Quality Control Analysis", "In manufacturing, maintaining high-quality standards is critical. When producing widgets, even a small defect rate can significantly impact customer satisfaction, warranty costs, and brand reputation. For a factory where widgets have a 3% defect rate, quality inspectors must assess the likelihood of defects in sampled batches to ensure compliance and drive improvements.", "The Scenario: Sampling with Precision", "Imagine a quality inspector randomly selecting 150 widgets from a large production line. Given a 3% defect rate, this situation provides an ideal example of binomial probability—a key concept in quality control and statistical analysis.", "### What Is a Binomial Distribution?", "The binomial distribution models the number of successes (in this case, defective widgets) in a fixed number of independent trials (sampled widgets), each with two possible outcomes: defective or non-defective.", "- Number of trials (n): 150\n- Probability of success (defect, p): 0.03\n- Number of successes (k): 5 (uses the binomial probability formula)", "### The Binomial Probability Formula", "The probability of exactly k defective widgets in n sampled units is:", "[\nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "Where:\n- $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient\n- $p = 0.03$ (probability a widget is defective)\n- $1 - p = 0.97$ (probability a widget is non-defective)", "### Applying the Values", "We want:", "[\nP(X = 5) = \binom{150}{5} (0.03)^5 (0.97)^{145}\n]", "Calculating this requires step-by-step computation:", "1. Binomial coefficient $\binom{150}{5}$:\n[\n\binom{150}{5} = \frac{150!}{5! \cdot 145!} \approx 591,600,000 \quad \ ext{(approximate via calculator or software)}\n]", "2. Defective part: $(0.03)^5 = 2.43 \ imes 10^{-9}$", "3. Non-defective part: $(0.97)^{145} \approx 0.0112$", "Combining:", "[\nP(X=5) \approx 591,!600,!000 \ imes 2.43 \ imes 10^{-9} \ imes 0.0112 \approx 0.0161\n]", "### Final Probability", "The probability that exactly 5 widgets are defective in a random sample of 150, given a 3% defect rate, is approximately 1.61%.", "---", "### Why This Matters in Quality Control", "Understanding such probabilities helps inspectors and managers:", "- Set realistic expectations for defect levels in samples\n- Estimate confidence intervals for quality metrics\n- Plan statistical tolerance limits and acceptance criteria\n- Justify inspection frequency and sample sizes", "For the 3% defect rate scenario, while 5 defects represent a minor fluctuation, knowing these probabilities enables data-driven decisions to maintain high-quality production standards.", "---", "Key Takeaways:", "- With a 3% defect rate, even moderate sample sizes reveal meaningful statistical insights.\n- Binomial models capture defect occurrences reliably in discrete trials.\n- Probability calculations empower proactive quality management and inspections.", "This approach exemplifies how statistical thinking strengthens manufacturing quality assurance, ensuring products meet expectations time after time."]