\( P(X=1) = \binom{8}{1} (0.12)^1 (0.88)^7 = 8 \times 0.12 \times 0.лия 0.3937 \approx 8 \times 0.12 \times 0.3937 \approx 0.3782 \)

["# Calculating ( P(X=1) ) Using the Binomial Distribution: A Step-by-Step Guide", "In probability theory and statistics, the binomial distribution is a powerful tool for modeling experiments with a fixed number of independent trials, each having two possible outcomes — success or failure. One common question in applied probability is calculating the probability of exactly ( k ) successes in ( n ) trials, denoted as ( P(X = k) ). This article walks you through the precise calculation of ( P(X = 1) ) using the binomial probability formula, using ( n = 8 ), ( p = 0.12 ), and ( q = 1 - p = 0.88 ).", "---", "## Understanding the Binomial Probability Formula", "The probability of getting exactly ( k ) successes in ( n ) independent trials is given by:", "[\nP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} = \binom{n}{k} p^k q^{n - k}\n]", "Where:\n- ( \binom{n}{k} ) is the binomial coefficient — the number of ways to choose ( k ) successes from ( n ) trials,\n- ( p ) is the probability of success on one trial,\n- ( q = 1 - p ) is the probability of failure,\n- ( X ) is a binomial random variable representing the number of successes.", "---", "## Applying the Formula to ( P(X = 1) ) in This Context", "Given:\n- ( n = 8 )\n- ( k = 1 )\n- ( p = 0.12 )\n- ( q = 1 - 0.12 = 0.88 )", "Plug into the formula:", "[\nP(X = 1) = \binom{8}{1} (0.12)^1 (0.88)^7\n]", "---", "## Step-by-Step Calculation", "1. Binomial Coefficient\n Compute ( \binom{8}{1} ):\n [\n \binom{8}{1} = \frac{8!}{1!(8 - 1)!} = \frac{8!}{1! \cdot 7!} = 8\n ]", "2. Success Term\n ( p^1 = (0.12)^1 = 0.12 )", "3. Failure Term\n ( q^{n - k} = (0.88)^7 )\n Calculate ( (0.88)^7 ):\n Using logarithms or a calculator:\n [\n 0.88^7 \approx 0.3937\n ]", "4. Multiply All Components\n Now combine:\n [\n P(X = 1) = 8 \ imes 0.12 \ imes 0.3937\n ]", "First compute:\n [\n 8 \ imes 0.12 = 0.96\n ]\n Then:\n [\n 0.96 \ imes 0.3937 \approx 0.3782\n ]", "---", "## Final Result", "[\nP(X = 1) = 8 \ imes 0.12 \ imes 0.3937 \approx 0.3782\n]", "So, the probability of exactly one success in 8 independent trials with a success probability of 12% per trial is approximately 0.3782, or 37.82%.", "---", "## Why This Calculation Matters", "This formula is essential in fields such as genetics, quality control, marketing analytics, and risk assessment where events occur repeatedly under consistent conditions. Understanding how to compute binomial probabilities helps in making data-driven decisions, assessing risks, and modeling real-world scenarios.", "---", "## Summary", "- The binomial probability ( P(X = 1) ) for ( n = 8 ), ( p = 0.12 ), is calculated using ( \binom{8}{1} (0.12)^1 (0.88)^7 ).\n- The value approximates to 0.3782 or 37.82%.\n- Use this method whenever you need to find the likelihood of a precise number of successes in fixed independent trials.", "Mastering this calculation empowers you to confidently work with discrete probability distributions in both academic and professional contexts."]