Question: A GIS analyst is labeling 10 urban zones with binary codes of length 4 (each digit 0 or 1). What is the probability that exactly three of the zones receive a code with exactly two 1s? - Parker Core Knowledge

Understanding the Context

In the age of geographic information systems (GIS), assigning consistent, unique codes to urban zones supports everything from emergency response routing to resource allocation and infrastructure planning. When cities label zones using 4-digit binary codes (each digit 0 or 1), each zone gains a unique identifier composed of only two active “1” markers — a compact yet informative system. Understanding the statistical behavior of these codes, such as how likely it is that three of ten zones end up with exactly two 1s, provides valuable insight into encoding efficiency and spatial data design. This knowledge helps planners minimize errors, avoid overlap, and optimize digital infrastructure crucial for smart urban growth.

The Code Probability: How Many Zones Get Two ‘1’s?

Each 4-digit binary code consists of four independent positions, each either 0 or 1. We define a “successful” code as one with exactly two 1s — a common structure in balanced binary systems. To find the chance that exactly three out of ten randomly labeled zones meet this criterion, we rely on binomial probability.

There are (2^4 = 16) total possible 4-digit binary codes. The number of codes with exactly two 1s is given by the combination ( \binom{4}{2} = 6 ). So, the probability a single zone gets a code with exactly two 1s is ( \frac{6}{16} = \frac{3}{8} = 0.375 ). With 10 zones labeled independently, this situation follows a binomial distribution with (n = 10), (p = 0.375), asking: what’s the chance exactly three zones qualify?

Key Insights

Using the binomial formula:

[ P(X = 3) = \binom{10}{3} \cdot (0.375)^3 \cdot (0.625)^7 ]

Calculating step-by-step:

• ( \binom{10}{3} = 120 )
• ( (0.375)^3 = 0.05273 )
• ( (0.625)^7 \approx 0.03755 )

Multiplying:

Final Thoughts

[ P(X = 3) \approx 120 \cdot 0.05273 \cdot 0.03755 \approx 0.2377 ]

So, the probability is about 23.77% — a moderate likelihood that exactly one in four zone codes aligns with the two-1s pattern. This insight underscores how rare yet measurable such configurations are, informing urban data design choices.

Broader Implications for GIS and Urban Analytics

Understanding these odds matters beyond probability — it feeds into smarter code design, better analytical workflows, and enhanced spatial data reliability. When analysts label zones with binary codes, they balance uniqueness, readability, and statistical stability. Having exactly three of ten zones land on a two-1s pattern may influence decisions about balancing code sets, reducing overlap, and improving data consistency. This approach supports scalable urban modeling, particularly as cities increasingly rely on systematic spatial tagging for AI-driven planning, asset tracking, and real-time analytics.

Common Questions About Zone Code Probabilities

Q: Why focus on exactly two 1s in a 4-digit binary code?
A: This count from two active markers helps define zones clearly while avoiding ambiguous inactivated states, offering efficient spatial categorization.