This is a binomial probability: n = 10, p = 0.8, find P(X ≥ 8) = P(X=8) + P(X=9) + P(X=10).

["# Understanding Binomial Probability: Calculating P(X ≥ 8) with n = 10 and p = 0.8", "In statistics, binomial probability is a fundamental concept used when dealing with a fixed number of independent trials, each with two possible outcomes: success (definitively counted) or failure. One common application involves calculating the probability of achieving a minimum number of successes — for instance, “what is the probability of getting at least 8 successes out of 10 trials?” This article explains how to compute ( P(X \geq 8) ) for a binomial distribution with ( n = 10 ) trials and success probability ( p = 0.8 ), using the formula:", "[\nP(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)\n]", "## What is a Binomial Distribution?", "The binomial distribution models the number of successes ( X ) in ( n ) independent Bernoulli trials, where each trial has a constant success probability ( p ). The probability of exactly ( k ) successes is given by:", "[\nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "- ( n = 10 ): total number of trials\n- ( p = 0.8 ): probability of success on any trial\n- ( 1 - p = 0.2 ): probability of failure\n- ( k ): number of successes (here, ( k = 8, 9, 10 ))", "Because we want ( P(X \geq 8) ), we sum the probabilities for exactly 8, 9, and 10 successes.", "## Step-by-Step Calculation of P(X ≥ 8)", "### 1. Calculate ( P(X = 8) )", "[\nP(X = 8) = \binom{10}{8} (0.8)^8 (0.2)^2\n]", "[\n\binom{10}{8} = \binom{10}{2} = \frac{10 \ imes 9}{2 \ imes 1} = 45\n]", "[\nP(X = 8) = 45 \ imes (0.8)^8 \ imes (0.2)^2\n]", "Calculate powers:\n( (0.8)^8 \approx 0.1678 )\n( (0.2)^2 = 0.04 )", "[\nP(X = 8) \approx 45 \ imes 0.1678 \ imes 0.04 \approx 45 \ imes 0.006712 = 0.3020\n]", "---", "### 2. Calculate ( P(X = 9) )", "[\nP(X = 9) = \binom{10}{9} (0.8)^9 (0.2)^1\n]", "[\n\binom{10}{9} = 10\n]", "[\nP(X = 9) = 10 \ imes (0.8)^9 \ imes 0.2\n]", "( (0.8)^9 \approx 0.1342 )\n[\nP(X = 9) \approx 10 \ imes 0.1342 \ imes 0.2 = 10 \ imes 0.02684 = 0.2684\n]", "---", "### 3. Calculate ( P(X = 10) )", "[\nP(X = 10) = \binom{10}{10} (0.8)^{10} (0.2)^0 = 1 \ imes (0.8)^{10} \ imes 1\n]", "( (0.8)^{10} \approx 0.1074 )\n[\nP(X = 10) \approx 0.1074\n]", "---", "## Summing the Probabilities", "[\nP(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10) \n\approx 0.3020 + 0.2684 + 0.1074 = 0.6778\n]", "---", "## Conclusion", "The probability of achieving at least 8 successes in 10 independent trials with a success rate of 80% is approximately 0.6778, or 67.78%. This calculation is a clear example of applying the binomial probability formula: summing individual probabilities ( P(X = k) ) for ( k = 8, 9, 10 ).", "Understanding binomial distributions helps in fields ranging from quality control and finance to medical trials and marketing analytics — wherever discrete trials with success/failure outcomes occur.", "If you want to compute binomial probabilities quickly, statistical software, calculators, or online tools can automate these calculations. But mastering the formula and logic behind them is essential for sound statistical reasoning.", "---", "Keywords: binomial probability, P(X ≥ 8), n = 10, p = 0.8, binomial distribution, probability calculations, statistical formula, success probability, binomial computation, statistic basics", "---", "Ready to calculate your own binomial probabilities? Use the formula ( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ) and sum for your desired range — your data’s potential outcomes are just a computation away!"]