Question:** A statistician is analyzing a discrete uniform distribution over integers from 1 to 84. What is the probability that a randomly selected integer from this range is a multiple of both 7 and 4?
["Understanding the Probability of a Number Being a Multiple of Both 7 and 4 in a Discrete Uniform Distribution", "When working with statistical analysis, understanding how to calculate probabilities over discrete uniform distributions is essential. In this article, we explore a specific case: determining the probability that a randomly selected integer from 1 to 84 is a multiple of both 4 and 7.", "First, recall that a discrete uniform distribution means every integer in the specified range has an equal chance of being selected. Here, the range is the integers from 1 to 84.", "### Step 1: Identify the total possible outcomes\nThe total number of possible outcomes is the number of integers in the range:", "[\n84 - 1 + 1 = 84\n]", "So, there are 84 possible integers.", "### Step 2: Determine the condition — multiples of both 4 and 7", "A number that is a multiple of both 4 and 7 must be a multiple of their least common multiple (LCM). Since 4 and 7 are coprime (they share no common factors other than 1), their LCM is:", "[\n\ ext{LCM}(4, 7) = 4 \ imes 7 = 28\n]", "Thus, we seek integers in the range [1, 84] that are multiples of 28.", "### Step 3: Find all multiples of 28 between 1 and 84", "List the multiples of 28 within the range:", "[\n28, , 56, , 84\n]", "We verify:\n- (28 \ imes 1 = 28)\n- (28 \ imes 2 = 56)\n- (28 \ imes 3 = 84)\n- (28 \ imes 4 = 112 > 84) → too large", "So, there are exactly 3 such numbers.", "### Step 4: Compute the probability", "Since the distribution is uniform, the probability is the number of favorable outcomes divided by the total outcomes:", "[\nP = \frac{\ ext{Number of multiples of both 4 and 7}}{\ ext{Total numbers from 1 to 84}} = \frac{3}{84} = \frac{1}{28}\n]", "### Conclusion", "The probability that a randomly selected integer from 1 to 84 is a multiple of both 4 and 7 is:", "[\n\boxed{\frac{1}{28}}\n]", "This example highlights how understanding least common multiples and basic counting within uniform distributions enables accurate probability calculations — a core skill for statisticians and data analysts."]
