#### 19.2Question: A civil engineer is designing a stormwater management system and randomly selects three integers between 0 and 100 inclusive to represent rainfall intensities. What is the probability that the product of these three integers is divisible by 5?

["Probability That the Product of Three Random Integers (0 to 100) Is Divisible by 5: A Civil Engineering Perspective", "When designing stormwater management systems, civil engineers often analyze rainfall patterns to ensure infrastructure can handle extreme weather. A key part of this analysis involves modeling rainfall intensities. Imagine the engineer randomly selects three integers between 0 and 100 inclusive—each representing a potential rainfall intensity. But a critical question arises: What is the probability that the product of these three numbers is divisible by 5?", "Understanding this probability helps engineers estimate worst-case scenarios and design effective drainage systems resilient to heavy rainfall集中 on divisibility by 5 due to its mathematical significance and relevance to environmental modeling.", "---", "### Why Focus on Divisibility by 5?", "A number’s divisibility by 5 depends solely on whether it includes a factor of 5. For the product of three numbers to be divisible by 5, at least one of the selected integers must be divisible by 5. Conversely, the product is not divisible by 5 only if none of the three integers is divisible by 5.", "---", "### Step 1: Total Possible Choices", "Each integer ranges from 0 to 100 inclusive—102 possible values (because 100 − 0 + 1 = 101 → wait, correction: 100 - 0 + 1 = 101 integers total).", "However, in many standardized mathematical contexts—especially when selecting uniformly at random—[0, 100] inclusive is taken to mean 102 values ([0, 1, 2, ..., 100]). But engineers modeling rainfall might include zero (representing no rainfall), so we assume inclusive range [0, 100] with 101 integers.", "Let’s clarify:\n- Total integers: 0, 1, 2, ..., 100 → 101 numbers\n- Among these, integers divisible by 5 are:\n 0, 5, 10, ..., 100 → this is an arithmetic sequence with difference 5.\n Number of such values = (100 − 0)/5 + 1 = 21", "So, 21 numbers between 0 and 100 inclusive are divisible by 5, and 80 numbers are not divisible by 5.", "---", "### Step 2: Total Number of Possible Triples", "Each of the three integers is chosen independently from 101 options.\nSo, total number of possible triples:\n[\n101^3 = 1,030,301\n]", "---", "### Step 3: Number of Triples Where Product Is Not Divisible by 5", "This happens when all three integers are NOT divisible by 5.\nNumber of non-divisible-by-5 choices: 101 − 21 = 80", "So, number of such triples:\n[\n80^3 = 512,000\n]", "---", "### Step 4: Number of Triples Where Product Is Divisible by 5", "We subtract to get favorable cases:\n[\n101^3 - 80^3 = 1,030,301 - 512,000 = 518,301\n]", "---", "### Step 5: Compute the Probability", "Probability = Favorable outcomes / Total outcomes\n[\nP = \frac{518,301}{1,030,301}\n]", "This fraction is already simplified (check via GCD if needed), but we can approximate:\n[\nP \approx 0.5031 \quad \ ext{or} \quad 50.31%\n]", "Interestingly, since there are 21 out of 101 integers divisible by 5 (~20.8%), the probability is slightly above 1/2 — because three independent draws increase the chance that at least one hits a multiple of 5.", "---", "### Engineering Insight: Why This Matters", "Civil engineers use such probabilities to model rare but high-impact events like intense storms. A product divisible by 5 ensures that rainfall intensity includes a "key threshold" value—critical when correlating rainfall with flood thresholds, pipe flow capacities, or soil saturation levels. Accurate probabilistic modeling supports resilient infrastructure design.", "---", "### Final Answer: The probability that the product of three randomly selected integers from 0 to 100 is divisible by 5 is:", "[\n\boxed{\frac{518301}{1030301}} \approx 0.5031\n]", "This means roughly 50.3% chance the combined rainfall intensity (as modeled by the product) hits a divisibility threshold relevant to stormwater planning—information vital for routing, storage, and overflow design.", "---", "Keywords: civil engineering, stormwater management, rainfall modeling, probability, product divisible by 5, infrastructure design, 0 to 100 integers, random selection, flood threshold, civil infrastructure, environmental modeling."]