How Many Drake 2-Bs Were Ever Made?
Counting Boatanchors with the German Tank Problem
I became interested in this question when Bill (N2CQR) was thinking out loud during an episode of the SolderSmoke podcast. He wondered how many Drake 2-B receivers the R.L. Drake Company had ever actually built, a question nobody at Drake is around to answer. The 2-B was a triple-conversion tube receiver without a crystal filter. Its final IF is 50 kHz (quite low!), and it employed an LC filter instead. It sold from roughly 1961 to 1966, a beloved boatanchor that still turns up at hamfests, but Drake apparently never published production totals.
At the time, I sent Bill an email about this question: There's a way to answer that, I told him — a lovely piece of wartime statistics called the German tank problem. However, it needed data, and I'd need his help. I asked Bill to ask his listeners to send in the serial numbers stamped on the backs of their 2-Bs. I could turn this data into an estimate. He put out the call on the next show, and the numbers came in via blog comments, emails, tweets, and Facebook messages. Bill even joked about carrier pigeons. When enough had piled up, he passed them on to me to run an analysis.
What follows is how that worked, and it's a great excuse to look at the same question two different ways: the frequentist way and the Bayesian way.
The wartime version
During World War II, the Allies wanted to know how many tanks Germany was producing. As it turns out, spies and aerial reconnaissance gave wildly inflated guesses. Statisticians took a different route. German tanks (and their gearboxes, wheels, and engines) carried sequential serial numbers, and every now and then, the Allies captured or destroyed one and could read the number off it. From a handful of captured serial numbers, the statisticians wondered, "Could you estimate the total?"
You can, and remarkably well. The post-war analysis was remarkable. For one stretch of the war, conventional intelligence estimated about 1,400 tanks a month, the serial-number method estimated around 245, and the captured German records put the true figure at 245. The spies were off by a factor of five, but the maths was off by essentially nothing.
Who actually came up with this? There's no single inventor. The technique was put to work by statisticians and economists in the Allied Economic Warfare Division — running serial and chassis numbers off captured German tanks, tyres, trucks, and even V-2 rockets — and the first published account is Richard Ruggles and Henry Brodie's 1947 paper in the Journal of the American Statistical Association. The mathematics that proves why the estimator is the best possible one came together separately around the same time: the Rao–Blackwell theorem (C. R. Rao, 1945; David Blackwell, 1947) and the Lehmann–Scheffé theorem (Erich Lehmann and Henry Scheffé, 1950) are exactly the tools that turn "the highest number you've seen" into the minimum-variance unbiased estimate we'll use below. The Bayesian view came later still, as a textbook reframing of the same puzzle.
The same idea has since been used to estimate Commodore 64 production, iPhone sales, and once Bill got his listeners going, Drake 2-B receivers.
The setup
Imagine the radios were numbered 1, 2, 3, … up to some unknown total N. We've observed a random sample of k of them, and the largest serial number in our sample is m. We want a best guess for N.
The whole thing rests on one assumption: that the serial numbers we happen to see are a fair, unbiased sample — that a radio with a high number is no more or less likely to reach us than one with a low number. For the tanks, this was about which ones got knocked out on the battlefield. For us it's about which owners happen to listen to a ham radio podcast and bother to walk over and read the back panel. More on whether that's fair below, so hold the thought.
The frequentist answer
Here's the intuition I gave Bill at the time, because it's not very maths-y and doesn't rely on years of statistical experience to follow.
Suppose you've seen exactly one radio, number m. What's your best guess for the total? Your single observation is, on average, going to land somewhere in the middle of the range, so a sensible guess is that the true top is about twice your observation — N ≈ 2m. This does assume there is no bias in your sampling of m. Now suppose you've seen two radios. The bigger of the two is probably closer to the top, so you'd scale up by less, roughly 1.5×. Three radios, fewer again. The more radios you see, the more confident you can be that your highest observation is already near the real ceiling, and your multiplier creeps down toward 1.
That intuition is expressed by this estimator, where the ^ on N means 'estimate of the true N':
N̂ = m (1 + 1/k) − 1
Read it as "the highest number you've seen, plus one average gap, minus a discrete-counting correction." If serials are evenly sprinkled through 1…N (unbiased), the typical spacing between them is about m/k, and the real top is, on average, one such gap above the biggest one you happened to catch. So you take your maximum and add a gap. With k = 1 it doubles your observation; as k grows it converges down onto m, which is just what it should do.
This is the minimum-variance unbiased estimator (MVUE) — among all unbiased methods, it's the one that wobbles the least from sample to sample. It is also gloriously simple: you need only two numbers, the largest serial and how many you've got.
The 2-B numbers (2011). In 2011, Bill's listeners produced 23 serial numbers. The highest was 12955. So m = 12955, k = 23:
N̂ = 12955 × (1 + 1/23) − 1 ≈ 13,517
So at the time, we concluded that the highest 2-B serial number ever stamped was probably around 13,500.
But the serials don't start at 1
But there was a complication I flagged at the time and couldn't fully close: Drake didn't start at serial number 1. Chasing this down later, the consensus among 2-B collectors, pieced together on SolderSmoke itself, is that the first 2-B had serial number 2000 and came from the first production run in April 1961.**
Here's the part I find genuinely amazing, elegant, and satisfying - it's one of the reasons I really love maths. The lowest serial number anyone reported to us was 2008 — just eight radios above the suggested floor of 2000. We didn't assume that floor to make the numbers work. It was proposed independently by collectors reasoning about Drake's production history, and our data completely reinforced this pubic impression. And the next-lowest numbers reinforce it further: 2046, 2258, 2532, all sitting just above 2000 and none below it. If the production line had really begun at 1, or at 500, or at 5000, that clustering would be an extraordinary coincidence. Instead, it's exactly the fingerprint a floor of 2000 would leave. Two independent lines of evidence, the collectors' history and the serial numbers themselves, agreeing to within eight units is about as good as this kind of amateur archaeological mathematics gets.
If the line runs from about 2000 to about 13,500, then the number of radios actually built is the length of that interval, not the top of it:
units built ≈ 13,500 − 2,000 ≈ 11,500
You can do this more cleanly by shifting every serial down by 1999 so the first radio becomes "number 1," applying the formula, and reading off the count directly. That gives roughly 11,400 units from the 2011 data. So let's say 11,5000 Drake 2-Bs were built.
(If you're feeling rigorous, you can also drop the assumption that the start is known and estimate both ends from the smallest and largest serials at once. That nudges the count up to about 11,900. The fact that "assume the floor is 2000" and "estimate the floor from the data" land within a few hundred of each other is itself reassuring. It means 2000 is a believable floor.)
The Bayesian answer
The frequentist gives you a single best number. The Bayesian approach asks a subtly and not so subtly different question: given the data, what's the whole distribution of plausible totals, and how sure should we be? I prefer this level of thinking, although it makes estimation more complex to perform and interpret.
For the Bayesian approach, you start with a prior guess, which is a distribution of possible answers, not a single answer, before seeing any serials. Our prior is, 'suppose every possible total N as equally believable' (a flat prior). Then ask, for each candidate N, how probable our actual sample was. The key fact: if there are really N radios and we draw k of them, the chance that our particular set fits inside 1…N is governed by how many ways k radios can be chosen from N, which is the binomial coefficient C(N, k). Bigger N means more ways to have drawn other radios instead, so larger totals are penalized. The posterior probability of each N is proportional to 1 / C(N, k)for every N at least as large as the maximum we observed.
That posterior has a long tail to the right — it rules out anything below m hard (we've seen m, so the total can't be smaller), but it allows ever-larger totals with steadily shrinking probability. The mean of the estimator (N̂) has a tidy closed form:
N̂_Bayes = (m − 1)(k − 1) / (k − 2) (for k ≥ 3)
Like the frequentist formula, it's "your maximum plus a bit," but because the posterior assigns large values equal probability, that "bit" is slightly larger. The two methods agree on the shape of the answer and differ only in temperament. The frequentist hands you a point and stops; the Bayesian hands you a point and an honest sense of the spread of possible answers. This is why I love Bayes in modeling and answering questions like this.
The 2-B numbers, Bayesian (2011), counting from a floor of 2000:
- Posterior-mean estimate: ≈ 11,477 units built
- 95% credible interval: roughly 10,970 to 12,950 units
In plain words: given 23 serials, we'd have been about 95% confident that Drake built somewhere between eleven thousand and thirteen thousand 2-Bs — and our single best guess, whichever school you belong to, sits right around eleven-and-a-half thousand. But Bayes gives you a range to consider, and that tightness of range and shape of the density of that range gives you so much more than just the mean estimator. So much more to think about. Best-case scenarios. Worst-case scenarios. That sort of thing. It doesn't leave you with just a single number which canbecome too much of a focus when answering questions.
Frequentist vs Bayesian, side by side
It's worth dwelling on why they nearly agree here, because the German tank problem is one of the cleanest places to see the two statistical philosophies meet.
The frequentist estimator is built to be unbiased: average it over many imaginary repetitions of the experiment, and it lands on 'the truth'. It is silent about any one experiment's uncertainty unless you separately work out its variance.
The Bayesian estimator starts from a prior and reports a full posterior distribution after seeing data, from which a credible interval falls out for free. With a flat prior and a decent sample, the two philosophies converge, which is exactly what we see: 13,517 (frequentist top serial) versus a Bayesian posterior implying a top serial around 13,476. The daylight between them is smaller than the uncertainty either one carries, so for practical purposes, the 2-B answer is robust to the method you use here. That robustness is the real result. When two different statistical worldviews, fed the same modest pile of hamfest serial numbers, both point at "about 11,500 radios," you can believe the number. But again, should we 'believe' in a number, or should we consider a credible range? I beleive we should consider a credible range.
A 2026 refresh
Here's something fun about leaving a blog post open for fifteen years: data keeps arriving. People are still finding this old SolderSmoke thread, walking over to their 2-B, and adding their serial number in the comments. Since the original analysis, eight more have come in — 2046, 2258, 2532, 4226, 6541, 7856, 10206, and 11093. The last result came in 2021.
None of them exceeds 12955, so our maximum observation is unchanged. But more samples sharpen the estimate. Re-running everything with all 31 serials:
| Quantity | 2011 (k = 23) | 2026 (k = 31) |
|---|---|---|
| Highest serial observed (m) | 12,955 | 12,955 |
| MVUE top serial (start = 1) | 13,517 | 13,372 |
| Units built (floor ≈ 2000) | ~11,431 | ~11,308 |
| Bayesian posterior-mean units | ~11,477 | ~11,333 |
| Bayesian 95% credible interval | 10,968 – 12,953 | 10,965 – 12,387 |
Our estimate barely moves. It is still about 11,300 to 11,500 radios. But look at the credible interval. The lower bound is rock-solid (it's pinned just above the highest serial we've seen), while the upper bound has pulled in from ~12,950 to ~12,390. Eight extra data points didn't change where we think the answer is; they changed how confidently we can rule out the possibility that there's a big stash of high-numbered 2-Bs we never sampled. That's the German tank problem showing off its best.
Play with the numbers yourself
The lovely thing about the German tank problem is that the estimate depends on only three things: the highest serial you've seen, how many radios you've sampled, and where the production numbers start. So you can recompute the whole analysis live. Try the 2011 and 2026 presets, or move the starting serial off 2000 and see how directly it shifts the count. The curve is the Bayesian posterior over the total number built; the dashed vertical line is the frequentist estimate, and the solid vertical line is the Bayesian mean. The height of the curve is a genuine probability: it represents the chance that the true total lies within roughly 100 radii of that point, peaking around 27% in our data. (Plotting it this way keeps the numbers readable — the probability of any single exact total is about a hundred times smaller, only a fraction of a percent, simply because the mass is spread across thousands of possible whole numbers.)
Two deliberate choices make the changes easy to see. First, the axes are fixed — the horizontal window and the 0–100% vertical scale don't rescale as you drag the sliders, so the curve genuinely grows taller and narrower in front of you instead of looking the same while the axis numbers quietly change underneath it. Second, a faint dashed reference curve stays pinned to the original 2011 sample (k = 23). As you increase the sample size, the solid curve climbs above the ghost and squeezes inward — a direct visual picture of more data buying more certainty.
Here's the thing to play with: drag the sample size k up high and watch two things happen at once. The credible-interval band squeezes inward, and the dashed (frequentist) and solid (Bayesian) lines slide together until they're almost on top of each other. That convergence, two different statistical philosophies agreeing more and more as the data grows, is exactly what happened in miniature when those eight extra serial numbers arrived between 2011 and now. More data doesn't just sharpen one estimate. It makes the rival methods agree.
For the record, here is the full set of serial numbers people sent in — every distinct one I could recover from the original SolderSmoke collection thread and its comments (31 in all, the highest being 12955):
2008, 2046, 2258, 2532, 2599, 4025, 4226, 4950, 5149, 5153, 5254, 6152, 6373, 6541, 7856, 8069, 8682, 8873, 9041, 9180, 9289, 10206, 10328, 10616, 11059, 11093, 11222, 11976, 12038, 12060, 12955
The widget starts loaded with exactly these numbers; hit a preset to compare the 2011 and 2026 snapshots, or "Load the real reported serials" to drop the full list back in.
German tank problem — Drake 2-B estimator
Adjust the inputs; everything recomputes instantly. Estimates use only the highest serial, the sample size, and the starting serial.
Advanced: paste your own serial numbers
When you use a pasted list, the sample size and highest serial are read straight from it. The “Load the real reported serials” button fills this in with the full list recovered from SolderSmoke.
| Sample size (k) | |
| Highest serial (m) | |
| Frequentist top serial (MVUE) | |
| Frequentist units built | |
| Bayesian 95% credible interval | |
| Bayesian best estimate (units built) |
The caveats (there are always caveats in statistics)
The whole edifice rests on that uniform-sampling assumption, and it's worth being honest about where that might fall apart:
- Survivorship. We only ever read serials off radios that still exist. If early 2-Bs were scrapped at a different rate from late ones, our sample would be skewed. The clustering of low numbers just above 2000 suggests that early units survive fine. This provides some verification of our assumptions.
- Self-selection. Our sample is "2-Bs owned by people who listen to SolderSmoke and chose to report." There's no obvious reason that should correlate with the serial number, which is precisely the condition this method needs, but it's an assumption, not 'proof'.
- Gaps and restarts. The maths assumes one unbroken run of numbers. If Drake skipped blocks, restarted sequences between production years, or shared a numbering scheme with another model, the estimate would drift. The clean fit to a single 2000-to-~13,500 line is encouraging but not conclusive.
None of these is fatal, and the fact that multiple methods and two independent statistical philosophies all converge on the same neighbourhood is the best evidence we have that the assumptions are roughly holding.
So, the answer
R.L. Drake built somewhere in the neighbourhood of 11,000 to 12,000 Model 2-B receivers, most likely right around 11,300, with the very last one carrying a serial number a little under 13,400. We arrived there not from a factory ledger but from a few dozen hams reading the backs of their radios and a formula the Allies used during World War II.
There's something deeply satisfying about that for me. A wartime statistical trick, a community of boatanchor enthusiasts, and a pile of serial numbers, adding up to a real answer about a sixty-year-old radio. Keep those numbers coming. Every new one tightens the interval a little more.
73 — Scott, K6AUS
Acknowledgments: this was a collaboration that began with an off-air note and Bill's willingness to run with it — thanks to Bill, N2CQR, for raising the question on the podcast, putting out the call, and keeping the thread alive all these years, and to every ham who walked over to their 2-B and reported what they found. The original collection and analysis live on the SolderSmoke Daily News blog (July and August 2011); the serial-number floor of 2000 comes from the collective sleuthing in the "Oldest Drake 2B?" thread.