Combined probability: \(\frac{7}{20} \times \frac{6}{19} = \frac{42}{380} = \frac{21}{190}\).

["# Understanding Combined Probability: How to Multiply Fractions and Simplify", "When learning about probability, one of the most essential concepts is combined probability — calculating the likelihood of two or more independent events happening together. A common formula used in such calculations is multiplying fractions to find the joint probability of independent events. In this article, we’ll explore combined probability using fractions, with a detailed look at the example:", "[\n\frac{7}{20} \ imes \frac{6}{19} = \frac{42}{380} = \frac{21}{190}\n]", "We’ll explain how this works, simplify fractions step-by-step, and clarify why this method is valuable in real-world applications like risk assessment, statistics, and decision-making.", "---", "## What Is Combined Probability?", "Combined probability refers to the probability that two or more independent events will all occur. If events are independent (the outcome of one doesn’t influence the other), we multiply their individual probabilities.", "Example:\nSuppose you flip a fair coin thrice (in independent trials), and ask:\n- What’s the probability of getting heads (probability = (\frac{1}{2})) three times in a row?\n- What’s the chance of drawing an ace from a deck, then rolling a 6 on a die?", "In both cases, because the events are independent, the combined probability is their product.", "---", "## Step-by-Step: Multiplying (\frac{7}{20} \ imes \frac{6}{19})", "Let’s analyze the example:\n[\n\frac{7}{20} \ imes \frac{6}{19}\n]", "### Step 1: Multiply the numerators", "Numerator multiplication is straightforward:\n[\n7 \ imes 6 = 42\n]", "### Step 2: Multiply the denominators", "[\n20 \ imes 19 = 380\n]", "### Step 3: Write the resulting fraction", "[\n\frac{42}{380}\n]", "### Step 4: Simplify the fraction", "To simplify (\frac{42}{380}), find the greatest common divisor (GCD) of 42 and 380.", "- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42\n- Factors of 380: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380\n- The largest common factor is 2", "Divide numerator and denominator by 2:\n[\n\frac{42 \div 2}{380 \div 2} = \frac{21}{190}\n]", "Final simplified form:\n[\n\boxed{\frac{21}{190}}\n]", "---", "## Why Is This Simplification Important?", "Fraction simplification ensures clarity and precision in probability reporting. Simplified fractions are easier to interpret, compare, and use in further calculations. In statistics and engineering, even small fractions matter—for instance, risk modeling or quality control—where accuracy minimizes errors.", "---", "## Real-World Applications of Combined Probability", "1. Insurance Risk Modeling\n Insurers calculate the joint probability of two events (e.g., a car accident AND theft) to price policies accurately.", "2. Medical Research\n Researchers examine whether two independent symptoms—like fever and cough—occur together with predictable frequency, aiding diagnosis.", "3. Engineering Reliability\n Engineers assess failure probabilities of independent system parts to enhance safety in aerospace, automotive, and construction.", "4. Game Theory and Strategic Planning\n In business or sports, understanding multiple dependent (but properly independent models) events enables better decision-making.", "---", "## Summary", "- Combining probabilities of independent events uses multiplication of fractions.\n- Example: (\frac{7}{20} \ imes \frac{6}{19} = \frac{42}{380} = \frac{21}{190}).\n- Simplification improves readability and accuracy.\n- Combined probability is vital across science, finance, engineering, and everyday decision-making.", "Mastering combined probability empowers you to analyze complex uncertain scenarios and make informed choices. Use fraction multiplication carefully — accuracy begins with the basics!", "---", "## Further Reading", "- Understanding Independent Events in Probability\n- Simplifying Fractions: Step-by-Step Guide\n- Applications of Probability in Real Life", "---", "Keywords: combined probability calculation, multiply fractions explained, simplify (\frac{7}{20} \ imes \frac{6}{19}), probability step-by-step, independent events probability, fraction simplification guide"]