Chapter 4
Probability and Uncertainty
Epigraphs
“Chance is perhaps the pseudonym of God when he does not want to sign.”
Anatole France
“Not everything that counts can be counted, and not everything that can be counted counts.”
Attributed to Einstein(einsteinIdeas)
Public administrators confront uncertainty far more often than certainty. Emergency calls fluctuate hour by hour, budgets shift with economic cycles, and neighborhood-level outcomes vary in ways that reflect both structural inequalities and unpredictable shocks (sampson2012great, chetty2016mto). Probability is the set of ideas and tools we use to formalize—rather than eliminate— uncertainty.
Opening Case: Dallas Flood Maps and the Meaning of a “100-Year Flood”
In 2023, the Dallas Morning News reported that updated Federal Emergency Management Agency (FEMA) flood-plain maps would redraw the boundaries of high-risk flood zones across Dallas County, adding thousands of properties to areas requiring federal flood insurance and removing others (dmnFloodMaps). The changes provoked sharp debate. Homeowners newly placed in flood zones faced insurance costs of several thousand dollars per year. Developers whose properties were removed from risk zones gained approval for projects that had previously been blocked. City engineers argued that the new maps better reflected current drainage infrastructure, while residents countered that the models failed to account for recent upstream development that had increased impervious surface area and runoff.
At the center of the controversy was a deceptively simple concept: the “100-year flood.” FEMA defines this as a flood event with a 1 percent annual probability of occurrence—not, as many residents assumed, a flood that happens once every hundred years. The distinction matters enormously. A 1 percent annual probability implies roughly a 26 percent chance of at least one such event over a 30-year mortgage. Misunderstanding this probability led both homeowners and elected officials to underestimate risk, while critics of the maps argued that even the 1 percent figure was too low given changing land use patterns and climate trends.
For an analyst, the flood map dispute is a case study in applied probability. Every element of this chapter—probability rules, conditional probability, rare events, and the assumptions built into distributional models—is visible in the public debate over which neighborhoods face flood risk, and why.
Guiding Questions
- What does it mean to say a neighborhood faces a "1\
- How do the assumptions built into FEMA's hydrological models---about rainfall, drainage capacity, and development patterns---shape the probability estimates on flood maps?
- If flood risk depends on both location and infrastructure investment, how would you express this relationship using conditional probability?
Why This Chapter Matters
Probability is foundational to every topic that follows: sampling, confidence intervals, hypothesis testing, regression, forecasting. Without a clear understanding of uncertainty, statistical inference collapses into guesswork.
In public administration, uncertainty is not merely an academic abstraction; it is an everyday operational constraint. City 311 call volumes spike unpredictably during storms or heat events (austin311open), and EMS incidents vary by hour, neighborhood, and season in ways that no deterministic staffing plan can fully anticipate. Demand for public services depends on economic conditions and community behavior, while budget revenues fluctuate with tax bases and business cycles. These sources of variation are not anomalies to be eliminated; they are inherent features of the systems administrators manage.
As Einav and Levin note, modern administrative datasets are shaped by data-generating rules, operational constraints, and behavioral responses—not idealized experiments (einav2014datarev). Probability provides structure for interpreting such data, allowing analysts to distinguish meaningful patterns from routine noise.
Briefing: In public administration, uncertainty is a feature of the system—not a flaw in the analyst.
The Concept of Uncertainty
Uncertainty arises when outcomes vary from one instance to the next. This variation might stem from natural randomness, such as the unpredictable arrival times of service requests, or from institutional processes like staffing levels and triage rules that change from shift to shift. Behavioral variation also plays a role: different neighborhoods report problems at different rates, and individual residents vary in how and when they contact government agencies. Environmental shocks—storms, power outages, seasonal temperature swings—add yet another layer of unpredictability. The goal of probability is not to predict perfect outcomes but to quantify these sources of uncertainty so that decisions can be made rationally.
Uncertainty appears in administrative datasets in two broad forms. The first is observable randomness: the number of pothole complaints varies day to day even when nothing about the road system has changed, daily water usage fluctuates with temperature, and EMS call rates shift by hour in patterns that are broadly predictable but never exact. Analysts can see this variation directly in the data and model it with probability distributions.
The second form is latent, structural randomness—uncertainty introduced by data-entry delays, categorization differences across staff members, and institutional processes that vary from one department to another. This type of variation is harder to detect because it is embedded in how data are collected and recorded rather than in the events themselves. Both forms reflect the institutional complexity highlighted in work on administrative data and neighborhood systems (einav2014datarev, sampson2012great).
Probability Basics
Probability assigns numerical values to uncertain events. Let $A$ be an event:
\[0 \leq P(A) \leq 1.\]If $P(A)=1$, the event is certain. If $P(A)=0$, it cannot happen. Most outcomes relevant to public administrators fall in between.
Three rules govern probability:
- **Non-negativity:** $P(A) \ge 0$
- **Normalization:** $P(\Omega) = 1$ for the full set of possible outcomes $\Omega$
- **Additivity:** If $A$ and $B$ are mutually exclusive,
These rules allow analysts to build structured models from operational uncertainty.
Briefing: Probability is not about predicting the next event—it’s about understanding the long-run pattern.
Random Variables
A random variable maps uncertain events onto numbers. In government operations, almost everything measured is a random variable: the number of 311 calls per hour, EMS response time, the count of overdue inspections, daily water consumption, and the time until a streetlight is repaired are all quantities whose values change unpredictably from one observation to the next. Formalizing these measures as random variables allows analysts to apply the tools of probability theory to operational questions.
A random variable $X$ may be discrete or continuous, and the distinction determines which probability models are appropriate. A discrete random variable takes on countable values, expressed as $X \in {0, 1, 2, \ldots}$. In the public sector, discrete random variables include the number of complaints received by a department in a day, the number of police incidents in a precinct during a shift, and the number of failed building inspections in a quarter. Because these variables count events, they are naturally modeled by distributions defined over non-negative integers.
A continuous random variable takes on values along a continuum, expressed as $X \in R}$. Examples from public administration include the time required to complete a service request, the dollar amount of property damage from a storm, and temperature-related demand for electricity. Continuous variables require density functions rather than simple probability tables, and their distributions often exhibit skewness and heavy tails that reflect the operational realities of government service delivery. Recognizing whether a variable is discrete or continuous is the first step toward selecting an appropriate probability model.
Probability Distributions
Random variables have probability distributions that describe how likely different values are.
For a discrete variable:
\[P(X = x_i) \ge 0, \quad \sum_i P(X = x_i) = 1.\]For a continuous variable, probabilities come from a density function $f(x)$:
\[P(a \le X \le b) = \int_{a}^{b} f(x) \, dx.\]Descriptive statistics tell us what happened in a sample; probability distributions tell us the patterns we expect in the long run.
This distinction mirrors the difference between neighborhood-level descriptive variation (sampson2012great) and national-level probabilistic mobility forecasts (chetty2016mto).
Briefing: Distributions translate uncertainty into structure.
Common Distributions in Public Administration
Some probability distributions recur so frequently in public-sector problems that they deserve special attention.
Poisson Distribution
The Poisson distribution models counts of independent events occurring over a fixed period of time or within a fixed region. Its probability mass function is:
\[P(X=k) = \lambda^k e^{-\lambda}}{k!}.\]The Poisson distribution appears throughout public administration because many operational processes involve counting discrete events that arrive more or less independently. A city department might use it to model the number of 311 complaints received per hour, allowing managers to estimate the probability of unusually high-volume periods and plan staffing accordingly. Emergency medical services rely on Poisson models to predict the number of calls per district per shift, which informs ambulance deployment and mutual-aid agreements. Traffic engineers apply the distribution to accident counts at intersections, identifying locations where observed crash frequencies exceed what the Poisson model predicts and thus warrant safety interventions. Even routine services such as library checkouts per day follow Poisson-like patterns when individual borrowing decisions are roughly independent of one another.
Worked Example: Poisson Probabilities for 311 Complaints
Suppose a city department receives an average of $\lambda = 5$ complaints per hour. What is the probability of receiving exactly 3 complaints in a given hour? And what is the probability of receiving 8 or more?
Step 1: $P(X = 3)$.
\[P(X=3) = 5^3 \, e^{-5}}{3!} = 125 \times 0.0067}{6} = 0.842}{6} = 0.140\]There is approximately a 14\
Step 2: $P(X \geq 8)$. Rather than computing infinitely many terms, use the complement: $P(X \geq 8) = 1 - P(X \leq 7)$. Summing the Poisson PMF for $k = 0, 1, \ldots, 7$:
\[P(X \leq 7) = 0.0067 + 0.0337 + 0.0842 + 0.1404 + 0.1755 + 0.1755 + 0.1462 + 0.1044 = 0.8666\] \[P(X \geq 8) = 1 - 0.8666 = 0.133\]About a 13\
[Figure — see PDF version]
Exponential Distribution
Describes time between events:
\[f(t) = \lambda e^{-\lambda t}, \qquad t \ge 0.\]Applications:
- time until next emergency call,
- time until next inspection failure,
- time between utility outages.
Normal Distribution
Although much administrative data are skewed (austin311open), some aggregated measures approximate normality due to the central limit theorem (Section below).
Heavy-Tailed Distributions
Completion times, repair delays, and public complaints often follow long right-tailed distributions because institutional bottlenecks produce extreme but informative cases (einav2014datarev).
Joint Probability and Dependence
Events interact. Joint probability measures the likelihood of multiple events happening together:
\[P(A \cap B).\]Events are:
- **Independent** if $P(A \cap B) = P(A)P(B)$,
- **Dependent** otherwise.
Dependence is common in public operations:
- Heat waves increase electricity outages and EMS calls simultaneously.
- Severe weather increases both flood complaints and traffic accidents.
- Staffing shortages increase the chance of delays and backlogs.
Such dependencies mirror the structural clustering documented in neighborhood-level research (sampson2012great).
Conditional Probability
Conditional probability focuses on how one event affects another:
\[P(A \mid B) = P(A \cap B)}{P(B)}.\]Interpretation in public administration:
- $P(delay} \mid weekend})$
- $P(high EMS volume} \mid heat advisory})$
- $P(missing data} \mid department})$ (gaoDataQuality)
Returning to the Case: The Dallas flood map case is a natural application of conditional probability. Residents want to know $P(flood} \mid in FEMA zone})$ versus $P(flood} \mid not in FEMA zone})$. But the probability also depends on infrastructure: $P(flood} \mid in zone, new drainage})$ may differ substantially from $P(flood} \mid in zone, aging infrastructure})$. The policy debate over flood maps is, at its core, a debate about which conditioning variables belong in the probability model.}
Worked Example: Flood Probability Over a Mortgage
A property in a FEMA 100-year flood zone faces a 1\
Step 1: Probability of no flood in a single year.
\[P(no flood in year } i) = 1 - 0.01 = 0.99\]Step 2: Probability of no flood in 30 years. Assuming independence across years:
\[P(no flood in 30 years}) = 0.99^{30} = 0.740\]Step 3: Complement.
\[P(at least one flood in 30 years}) = 1 - 0.740 = 0.260\]A 1\
Conditional patterns often reveal deep institutional insights—for example, that delays cluster in specific neighborhoods or during certain seasons.
Briefing: Conditional probability reveals hidden structure in operational systems.
Bayes’ Rule
Bayes’ Rule updates beliefs in light of new evidence:
\[P(A \mid B) = P(B \mid A) P(A)}{P(B)}.\]Examples:
- Updating outage risk when the weather forecast changes.
- Revising projected call volume after early-morning patterns are observed.
- Updating likely inspection failures given new information about facility age.
Bayesian reasoning mirrors real-time decision-making in emergency management, where agencies adjust plans dynamically.
Expectation and Variance
The expected value of a random variable $X$ is its long-run average:
\[E[X] = \sum_i x_i P(X=x_i) \quad (discrete)},\] \[E[X] = \int x f(x) \, dx \quad (continuous)}.\]Variance measures dispersion:
\[Var}(X) = E[(X - E[X])^2].\]These quantities are not just mathematical abstractions. They apply directly to:
- forecasting call volumes,
- budgeting under uncertainty,
- assessing workload variability,
- designing performance standards (ombPerfGuide).
High-variance neighborhoods often correspond to the structural inequalities documented in urban research (sampson2012great).
Law of Large Numbers
The Law of Large Numbers (LLN) states:
\[X}_n \to E[X] \quad as } n \to \infty.\]This explains why aggregated administrative metrics (monthly totals, yearly averages) are stable even when individual observations vary wildly.
Federal agencies such as the Census Bureau and the Bureau of Labor Statistics (BLS) rely on LLN when reporting economic and demographic indicators (uscensusACS, blsHandbook).
Briefing: The LLN explains why aggregate trends look orderly even when individual cases look chaotic.
Central Limit Theorem
The Central Limit Theorem (CLT) states that averages of many independent observations tend toward a normal distribution:
\[X} - E[X]}{\sigma/n}} \to N(0, 1).\]The CLT justifies why:
- budget forecasts use normal-theory confidence bands,
- performance measures use normal approximations,
- sampling distributions behave predictably.
But independence and identical distribution assumptions can fail in neighborhood-level processes (sampson2012great), which is why analysts must interpret CLT results carefully.
Briefing: The CLT makes inference possible—but only when the administrative context supports it.
Probability in Public Administration Practice
Forecasting
\[P(High Volume Tomorrow})\]Used by 311 call centres to schedule staffing during weather events.
Risk Assessment
\[P(Severe Delay} \mid Low Staffing})\]Helps identify operational vulnerabilities.
Equity Analysis
\[P(Delay} \mid Neighbourhood Characteristics})\]Connects descriptive disparities to structural probabilities, consistent with neighborhood effects research (sampson2012great,chetty2016mto).
Performance Standards
Agencies often set probability-based standards (e.g., 90
(ombPerfGuide).
Common Pitfalls
Assuming Independence
Administrative processes are rarely independent; weather, staffing, and neighborhood conditions create dependence structures.
Overinterpreting Short-Run Frequencies
Small samples produce deceptive probabilities.
Treating Rare Events as Impossible
Returning to the Case: Dallas homeowners frequently interpreted their “100-year flood zone” designation as meaning a flood would not happen in their lifetime. But a 1\
Ignoring Data-Generating Processes
Analysts must understand how data are collected (einav2014datarev, gaoDataQuality) before applying probability models.
Rare-but-consequential events (system outages, extreme delays, revenue collapses) matter disproportionately.
Briefing: Probability requires humility. Misapplied, it produces confident but empty claims.
Practice and Application
- Using 311 completion time data, estimate $P(Delay} > 7 days})$ and compare across neighborhoods (sampson2012great).
- Compute conditional probabilities such as $P(High Volume} \mid Rain})$ using hourly call data (austin311open).
- Simulate Poisson arrivals for EMS calls and compare simulation outcomes with real administrative data (dshsEMS).
- Use Bayes’ Rule to update the probability of a major outage given new weather alerts.
- Write a short memo explaining why LLN and CLT justify (or fail to justify) current departmental performance metrics.
- A Texas city experiences an average of 3 water main breaks per month. Using the Poisson distribution, compute (in Excel or by hand) the probability of (a) zero breaks in a given month, (b) exactly 5 breaks, and (c) 6 or more breaks. Show your work. What staffing implications does the answer to (c) have?
- Using the flood probability formula from this chapter, compute the probability of at least one major flood ($P = 0.02$ annually) over a 50-year planning horizon. Compare this to the 30-year mortgage calculation in the worked example. Write a one-paragraph memo to a city planner explaining why a "50-year flood" is not as rare as it sounds.
Sources for Examples in This Chapter
- City of Austin 311 open data portal (austin311open).
- Administrative data processes (einav2014datarev).
- Neighbourhood variation research (sampson2012great,chetty2016mto).
- Federal standards and measurement guidance (ombPerfGuide).
- Data reliability and reporting guidance (gaoDataQuality).
- ACS and BLS methodological documentation (uscensusACS,blsHandbook).
Transition to Chapter Five
Probability provides the conceptual foundation for statistical inference. In the next chapter, we use probability to evaluate whether differences in data are meaningful or simply due to chance. Chapter~chap:inference introduces that toolkit.