5Question: A data scientist uses a fair 12-sided die to simulate random trials. What is the probability that exactly one of three rolls results in a number divisible by 4, and the other two do not?
["Title: Probability of Exactly One Success in Three Rolls Using a Fair 12-Sided Die: Exactly One Divisible by 4", "---", "Understanding probability in practical scenarios like dice rolling can greatly benefit data scientists, especially when validating randomness or simulating experimental outcomes. One common question in probability theory is: What is the probability that exactly one of three rolls results in a number divisible by 4, using a fair 12-sided die?", "This article breaks down the solution step-by-step, integrating both combinatorial reasoning and probability fundamentals—essential tools in a data scientist’s toolkit.", "---", "### Understanding the Problem", "You roll a fair 12-sided die three times. The die faces are numbered from 1 to 12. We want the probability that exactly one roll shows a number divisible by 4, while the other two rolls show numbers not divisible by 4.", "---", "### Step 1: Identify Favorable Outcomes for Success and Failure", "First, determine how many numbers from 1 to 12 are divisible by 4:\nMultiples of 4 in this range: 4, 8, 12 → 3 favorable outcomes", "So, probability of rolling a number divisible by 4 on one roll:", "[\nP(\ ext{divisible by 4}) = \frac{3}{12} = \frac{1}{4} = 0.25\n]", "Thus, probability the number is not divisible by 4:", "[\nP(\ ext{not divisible by 4}) = 1 - \frac{1}{4} = \frac{3}{4} = 0.75\n]", "---", "### Step 2: Model the Experiment", "Each die roll is an independent Bernoulli trial:\n- Success (divisible by 4): ( p = \frac{1}{4} )\n- Failure (not divisible by 4): ( q = \frac{3}{4} )", "We perform 3 independent trials, and seek the probability of exactly one success.", "This follows a binomial distribution with parameters ( n = 3 ), ( p = \frac{1}{4} ), and exactly ( k = 1 ) success.", "The binomial probability formula is:", "[\nP(k \ ext{ successes}) = \binom{n}{k} p^k (1-p)^{n-k}\n]", "---", "### Step 3: Plug in Values", "[\nP(\ ext{exactly one divisible by 4}) = \binom{3}{1} \left(\frac{1}{4}\right)^1 \left(\frac{3}{4}\right)^2\n]", "Calculate each part:", "- ( \binom{3}{1} = 3 )\n- ( \left(\frac{1}{4}\right)^1 = \frac{1}{4} )\n- ( \left(\frac{3}{4}\right)^2 = \frac{9}{16} )", "Now compute:", "[\nP = 3 \ imes \frac{1}{4} \ imes \frac{9}{16} = \frac{27}{64}\n]", "---", "### Step 4: Final Answer and Interpretation", "The probability that exactly one of three rolls results in a number divisible by 4 (and the other two do not) is:", "[\n\boxed{\frac{27}{64}}\n]", "This result exemplifies how data scientists use probability models to analyze randomized processes—critical for validating simulations, designing experiments, or assessing algorithmic fairness.", "---", "### Why This Matters for Data Scientists", "Understanding such probabilities enables better modeling of stochastic systems, debugging random number generators, and interpreting results from Monte Carlo simulations. The fair die serves as a simple yet powerful abstraction of real-world randomness.", "---", "Key terms for SEO: data scientist, probability calculation, binomial distribution, fair 12-sided die, exactly one success, random trials simulation, count and probability, probability simulation", "---", "Keywords: probability of exactly one divisible by 4 in three rolls of a 12-sided fair die"]
