This is a binomial probability with $ n = 5 $, $ k = 3 $, and success probability $ p = \frac{1}{3} $.
["# Binomial Probability with ( n = 5 ), ( k = 3 ), and ( p = \frac{1}{3} )", "When analyzing outcomes involving repeated independent trials with two possible results, binomial probability provides a powerful mathematical model. This article explores the binomial probability scenario where we perform ( n = 5 ) trials, aim for exactly ( k = 3 ) successes, and each trial has a success probability of ( p = \frac{1}{3} ), with failure probability ( q = 1 - p = \frac{2}{3} ).", "---", "## Understanding the Binomial Probability Formula", "The binomial probability formula calculates the probability of obtaining exactly ( k ) successes in ( n ) independent trials, where each trial has a success probability ( p ). The formula is:", "[\nP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "In our case:\n- ( n = 5 )\n- ( k = 3 )\n- ( p = \frac{1}{3} )\n- ( q = 1 - p = \frac{2}{3} )", "---", "## Calculating the Binomial Coefficient", "The binomial coefficient ( \binom{n}{k} ) counts the number of ways to choose ( k ) successes out of ( n ) trials:", "[\n\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \ imes 4 \ imes 3!}{3! \ imes 2!} = \frac{20}{2} = 10\n]", "So, there are 10 different ways to achieve exactly 3 successes in 5 trials.", "---", "## Applying the Probability Values", "Substituting values into the binomial formula:", "[\nP(X = 3) = \binom{5}{3} \left(\frac{1}{3}\right)^3 \left(\frac{2}{3}\right)^{5-3}\n= 10 \cdot \left(\frac{1}{3}\right)^3 \cdot \left(\frac{2}{3}\right)^2\n]", "Calculate each part:", "- ( \left(\frac{1}{3}\right)^3 = \frac{1}{27} )\n- ( \left(\frac{2}{3}\right)^2 = \frac{4}{9} )", "Now multiply all components:", "[\nP(X = 3) = 10 \cdot \frac{1}{27} \cdot \frac{4}{9} = 10 \cdot \frac{4}{243} = \frac{40}{243}\n]", "---", "## Final Result", "The probability of obtaining exactly 3 successes in 5 trials with success probability ( \frac{1}{3} ) is:", "[\n\boxed{\frac{40}{243}} \approx 0.1646 \quad \ ext{(about 16.46%)}\n]", "---", "## Practical Applications", "This binomial model applies in diverse fields:\n- Quality control: Calculating the chance of exactly 3 defective items in 5 sampled products.\n- Medical trials: Assessing success rates in drug administration across multiple patients.\n- Marketing research: Estimating response rates when testing customer preferences with a small success likelihood.", "---", "Understanding binomial distributions empowers decision-making under uncertainty, offering precise calculations for common real-world scenarios involving repeated independent events.", "For further reading, explore variations such as cumulative binomial probability and how different success probabilities affect outcomes.", "---", "Keywords: binomial probability, n = 5, k = 3, p = 1/3, success probability, binomial coefficient, probability distribution,athacibability,statistics,examples"]
