During World War II, Allied intelligence needed to estimate the total production of German tanks, trucks, and other military equipment. Conventional intelligence methods — interrogation, aerial reconnaissance, spy networks — produced wildly variable estimates. But a team of statisticians at the British Ministry of Supply noticed that captured equipment bore sequential serial numbers. By applying statistical estimation to these serial numbers, they could infer the total number produced — and their estimates proved remarkably accurate.
The problem — estimating the maximum of a discrete uniform distribution from a sample — has become one of the most celebrated examples in mathematical statistics, precisely because Bayesian and frequentist methods give different answers with distinct interpretations, and because the real-world application provides a dramatic illustration of statistical reasoning in action.
The Setup
Suppose tanks are numbered 1, 2, …, N, where N is the unknown total production. We observe k serial numbers from captured tanks. Let m denote the maximum serial number observed. The goal is to estimate N.
Sample size: k (number of captured tanks)
Observed maximum: m = max(s₁, s₂, …, sₖ)
Likelihood P(max = m | N, k) = C(m−1, k−1) / C(N, k) for m ≤ N
The Frequentist Solution
The minimum variance unbiased estimator (MVUE) of N is:
Example k = 5 captured tanks, max serial number m = 60
N̂ = 60 × 6/5 − 1 = 72 − 1 = 71
Intuition The sample maximum m underestimates N. On average, you'd expect the gap
between m and N to be about m/k, so N̂ ≈ m + m/k.
This estimator has a beautifully intuitive interpretation. The sample partitions the range [1, N] into (k+1) intervals: below the smallest observed number, between each pair of consecutive observed numbers, and above the maximum. On average, these intervals are equal in length, so the expected gap above the maximum is N/(k+1). Adding this expected gap to m gives the estimator.
The Bayesian Solution
The Bayesian approach requires a prior distribution over N. Different priors lead to different posteriors, but the most common choices illustrate the range of Bayesian reasoning.
With Discrete Uniform Prior on N (1 to N_max) p(N | m, k) ∝ C(N, k)⁻¹ for N ≥ m
With Improper Prior p(N) ∝ 1/N p(N | m, k) ∝ C(N, k)⁻¹ / N for N ≥ m
Posterior Mean (Uniform Prior) E[N | m, k] = (m−1)(k−1) / (k−2) for k > 2
For k=5, m=60: the frequentist MVUE gives N-hat = 71; the Bayesian posterior mean with a uniform prior gives (59)(4)/(3) = 78.7; with the 1/N prior, the posterior mean is approximately 74. The estimates differ because they answer different questions. The frequentist estimator minimizes squared error on average over repeated sampling. The Bayesian estimate minimizes expected squared error under the posterior. Both are reasonable; they simply optimize different criteria.
The Historical Application
The statistical approach was applied to estimate production of various German military items: Panzer V (Panther) tanks, Tiger tanks, trucks, V-1 and V-2 rockets, and tires. The statisticians used not only the maximum serial number but also the spacing between observed serial numbers to refine their estimates.
Allied statisticians at the Ministry of Supply begin systematic collection of serial numbers from captured German equipment on the North African and Italian fronts.
Statistical estimates suggest German monthly tank production is around 250 — far lower than the conventional intelligence estimate of 1,400. Military planners are skeptical.
After the war, Albert Speer's production records reveal that actual German tank production was approximately 255 per month during the relevant period — strikingly close to the statistical estimate and far from the intelligence estimate.
"The statisticians were closer to the truth than any other intelligence source. Their analysis of serial numbers produced estimates that, when checked against captured German records, proved to be remarkably accurate." — Ruggles and Brodie, "An Empirical Approach to Economic Intelligence in World War II" (1947)
Mathematical Properties
Sufficient Statistic
The maximum m is a sufficient statistic for N — it captures all the information the sample contains about the population size. This follows from the factorization theorem: the likelihood depends on the data only through m (and the known sample size k).
Confidence Intervals
The distribution of the maximum, P(max = m | N, k) = C(m-1, k-1)/C(N, k), allows exact computation of frequentist confidence intervals for N. The 95% confidence interval is [m, N_upper], where N_upper is found by solving C(m-1, k-1)/C(N_upper, k) = 0.05.
Bayesian Credible Intervals
The Bayesian posterior p(N | m, k) directly provides credible intervals. The 95% highest posterior density (HPD) interval includes all values of N with posterior density above some threshold, chosen so that the total posterior probability in the interval is 0.95.
Modern Applications
The German tank problem is not merely a historical curiosity. The same statistical framework applies whenever items in a population carry sequential identifiers.
Software and Technology
Estimating the total number of software bugs from bug IDs, or the production volume of a competitor's product from observed serial numbers, are modern analogues. Analysts have estimated production runs of military aircraft, smartphones, and gaming consoles using serial number analysis.
Ecology
Capture-recapture methods for estimating animal population sizes are mathematically related. The Lincoln-Petersen estimator for mark-recapture studies has a similar structure to the German tank estimator.
The German tank problem appears in virtually every introductory Bayesian statistics course because it elegantly illustrates how the choice of prior affects the posterior, how Bayesian and frequentist approaches differ in their reasoning yet often produce similar practical answers, and how statistical thinking can extract profound conclusions from deceptively simple data. It remains one of the best examples for teaching the foundations of statistical inference.