P(X=8) = C(10,8) (0.8)^8 (0.2)^2 = 45 × 0.16777216 × 0.04 ≈ 0.30199
["Understanding P(X=8) in Binomial Probability: A Detailed Explanation of C(10,8)(0.8)^8(0.2)^2 ≈ 0.302", "In probability theory and statistics, binomial distributions are fundamental for modeling events with two possible outcomes (success/failure, 0/1, etc.). One key concept is calculating the probability of exactly ( k ) successes in ( n ) independent trials, commonly expressed as:", "[\nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "This formula applies perfectly in scenarios such as quality control, survey analysis, or any situation where outcomes are binary and repeated under identical conditions.", "---", "### Calculating ( P(X = 8) ) for Binomial Distribution", "Let’s explore the computation of ( P(X = 8) ) when:", "- Number of trials: ( n = 10 )\n- Number of successes: ( k = 8 )\n- Probability of success in one trial: ( p = 0.8 )\n- Probability of failure: ( 1 - p = 0.2 )", "Applying the binomial probability formula:", "[\nP(X = 8) = \binom{10}{8} (0.8)^8 (0.2)^2\n]", "First, compute the binomial coefficient:", "[\n\binom{10}{8} = \frac{10!}{8! \cdot (10-8)!} = \frac{10 \ imes 9}{2 \ imes 1} = 45\n]", "Next, evaluate the probability terms:", "[\n(0.8)^8 = 0.16777216 \quad \ ext{(exact value on calculator)}\n]", "[\n(0.2)^2 = 0.04\n]", "Now multiply all components together:", "[\nP(X = 8) = 45 \ imes 0.16777216 \ imes 0.04 = 45 \ imes 0.0067108864 \approx 0.30199\n]", "Thus,", "[\nP(X = 8) \approx 0.302\n]", "---", "### Why This Formula Matters", "This precise calculation is crucial in:", "- Predictive analytics: Estimating outcomes in repeated experiments.\n- Risk assessment: Calculating the likelihood of a set number of successes (e.g., defective items in a batch).\n- Decision-making: Informing strategies based on probabilistic forecasts.", "Understanding every component — from combinatorics to exponents — ensures accurate interpretation and application in real-world data science, engineering, and business contexts.", "---", "### Summary", "The expression ( P(X=8) = \binom{10}{8} (0.8)^8 (0.2)^2 \approx 0.302 ) clearly demonstrates the power of the binomial distribution. By combining combinations (counting favorable arrangements) with probabilistic powers, we quantify uncertainty in a structured, reliable way — a cornerstone of statistical modeling today.", "Keywords: binomial probability, P(X=8), C(10,8), (0.8)^8, (0.2)^2, probability calculation, statistical formula, quality control, data analysis, exponents in probability.", "---", "If you’re analyzing events with binary outcomes, mastering this formula equips you to compute precise probabilities and make data-driven decisions with confidence."]
