Question: A spatial analyst models city zones as regions, each either high-income or low-income. In a simulation, 7 zones are randomly assigned, with 4 high-income and 3 low-income. If 3 zones are selected at random for a detailed equity audit, what is the probability that all 3 zones are of different types?

Sun, 03 May 2026 03:08:53 +0000

**Why Zoning Equity Audits Are Sparking New Conversations in U.S. Cities**
A recent surge in urban planning discussions highlights growing public and policymaker interest in how income distribution shapes opportunity across neighborhoods. Simulations like selecting three randomly assigned city zones from a mix of high-income and low-income regions reveal hidden patterns in equity. With 7 zones overall—4 high-income and 3 low-income—equity audits aim to uncover disparities, especially when choosing diverse samples. Interest peaks when data-driven insights inform fairness in resource allocation, infrastructure investment, and community development. **Why This Simulation Matters for Urban Equity**
This question isn’t just academic—it reflects a rising trend toward transparency. As cities use simulation models to assess socioeconomic diversity, the chance of randomly picking a mix of key income zones offers key insights. It reveals how rare it is to encounter every type in a small sample—helpful for auditors seeking balanced representation. In mobile-RT ecosystems where users research urban policy or neighborhood change, such data supports informed civic engagement. **Understanding the Probability: A Step-by-Step Breakdown**
To find the chance that 3 randomly chosen zones are all different types, consider the only two types: high-income (H) and low-income (L). With 4 H and 3 L zones total, selecting 3 zones that include both types—and specifically one of each—is a nuanced probability. Since only two categories exist, “all different types” requires one H and two L, or two H and one L—but the phrasing of “different types” implies a mix: one high, one low, and one ambiguous. However, in this binary framework, ‘different types’ captures all non-identical distributions. We calculate the chance all zones reflect diversity within the sample, avoiding misleading terms linked to adult content. **How to Calculate the Probability**
Start by estimating the likelihood of selecting each zone type in three draws without replacement. The total ways to pick any 3 zones from 7 is C(7,3) = 35. To get a mixed sample—one H and two L or two H and one L—that includes both types—actually, only samples with at least one of each matter. But since all combinations include either H or L more than once only when all three match, the only mismatch is uniform. Instead, the clause “all different types” here refers to no single dominant type dominating—so we interpret it as a balanced selection. The real target is selecting one H and two L, or two H and one L—both reflect mixed typing—plus accounting for order and exact composition.
The accurate way: Probability all three zones represent both zones types, meaning neither all H nor all L.
Total ways: C(7,3) = 35
All H: C(4,3) = 4
All L: C(3,3) = 1
Mixed (H, L, and either): 35 − 4 − 1 = 30
Thus, probability all zones reflect mixed income types is 30/35 = 6/7 ≈ 85.7%, a number gaining traction in equity modeling and accessible to mobile users seeking clarity.