Solution: Using the binomial distribution with $n = 3$, $p = \frac{1}{4}$, and $k = 1$:

["# Using the Binomial Distribution with ( n = 3 ), ( p = \frac{1}{4} ), and ( k = 1 ): A Practical Solution for Probability Modeling", "Understanding probability distributions is essential in statistics, data science, and decision-making across various fields. One of the most widely used discrete distributions is the binomial distribution, ideal for modeling the number of successes in a fixed number of independent trials. In this article, we explore a concrete application: calculating the probability of exactly one success in three trials, where the probability of success on a single trial is ( p = \frac{1}{4} ). We’ll walk through the calculation using the binomial probability formula with ( n = 3 ), ( p = \frac{1}{4} ), and ( k = 1 ).", "---", "## What Is the Binomial Distribution?", "The binomial distribution models the number of successes ( k ) in ( n ) independent Bernoulli trials, each with success probability ( p ). Its probability mass function is given by:", "[\nP(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}\n]", "where:\n- ( n ) = number of trials\n- ( k ) = number of successes\n- ( p ) = probability of success on a single trial\n- ( \binom{n}{k} ) is the binomial coefficient, representing the number of ways to choose ( k ) successes from ( n ) trials\n- ( (1 - p) = q ) is the probability of failure", "---", "## Applying the Formula to Our Case", "We are given:\n- ( n = 3 ) (three independent experiments)\n- ( p = \frac{1}{4} ) (probability of success per trial)\n- ( k = 1 ) (exactly one success desired)", "### Step 1: Compute the binomial coefficient\n[\n\binom{3}{1} = \frac{3!}{1!(3 - 1)!} = \frac{6}{1 \ imes 2} = 3\n]", "### Step 2: Plug values into the formula\n[\nP(X = 1) = \binom{3}{1} \left(\frac{1}{4}\right)^1 \left(1 - \frac{1}{4}\right)^{3 - 1}\n= 3 \cdot \frac{1}{4} \cdot \left(\frac{3}{4}\right)^2\n]", "### Step 3: Simplify the expression\n[\n\left(\frac{3}{4}\right)^2 = \frac{9}{16}\n]\n[\nP(X = 1) = 3 \cdot \frac{1}{4} \cdot \frac{9}{16} = \frac{27}{64}\n]", "---", "## Final Answer", "The probability of getting exactly one success in three independent trials, each with a success probability of ( \frac{1}{4} ), is:", "[\n\boxed{\frac{27}{64}}\n]", "This result reflects the precise chance of one favorable outcome among three attempts under the specified conditions—making it a valuable outcome for risk analysis, quality control, and scientific experimentation.", "---", "## Why Use the Binomial Distribution with These Parameters?", "This particular setup is useful in numerous real-world contexts:", "- Medical trials: Estimating the chance of exactly one patient responding to treatment out of three administered cases.\n- Quality testing: Predicting how many defective items are found in a small sample when defect probability is known.\n- Marketing research: Modeling customer behavior such as exactly one out of three surveyed clients making a purchase.", "Using ( n = 3 ), ( p = \frac{1}{4} ), and ( k = 1 ) demonstrates a balanced scenario where theoretical probability translates directly into actionable insight.", "---", "## Conclusion", "The binomial distribution provides a powerful framework for analyzing discrete success events with fixed probabilities. By plugging in known values into ( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ), we confidently compute ( P(X = 1) = \frac{27}{64} ) for ( n = 3 ), ( p = \frac{1}{4} ), and ( k = 1 ). This approach not only reinforces foundational statistical concepts but also enables informed decision-making grounded in probability theory.", "If you’re working with similar problems or seeking clear, accurate probability calculations, mastering the binomial model equips you with a crucial tool for data-driven success."]