German tank problem In the statistical theory of estimation , the problem of estimating the maximum of a discrete uniform distribution from sampling without replacement , is known , in the English - speaking world , as the German tank problem , due to its application in World War II to the estimation of the number of German tanks . The analyses illustrate the difference between frequentist inference and Bayesian inference . Estimating the population maximum based on a " single " sample yields divergent results , while the estimation based on " multiple " samples is an instructive practical estimation question whose answer is simple but not obvious . Example . Suppose an intelligence officer has spotted tanks with serial numbers , 2 , 6 , 7 , and 14 , with the maximum observed serial number , . The unknown total number of tanks is called " N " . The formula for estimating the total number of tanks suggested by the frequentist approach outlined below is Whereas , the Bayesian analysis below yields ( primarily ) a probability mass function for the number of tanks from which we can estimate the number of tanks according to This distribution has positive skewness , related to the fact that there are at least 14 tanks . Historical problem . During the course of the war the Western Allies made sustained efforts to determine the extent of German productivity , and approached this in two major ways : conventional intelligence gathering and statistical estimation . In many cases statistical analysis substantially improved on conventional intelligence . In some cases conventional intelligence was used in conjunction with statistical methods , as was the case in estimation of Panther tank production just prior to D - Day . The allied command structure had thought the Panther tanks seen in Italy , with their high velocity , long barreled 75 mm / L70 guns , were an unusual heavy tank , and would only be seen in northern France in small numbers , much the same way as the Tiger I was seen in Tunisia . The US Army was confident that the Sherman tank would perform well against the German Panzer Mk III and IV tanks which they expected to meet . Shortly before D - Day , rumors indicated that large numbers of Panther tanks were being used in the panzer divisions . To ascertain if this were true the Allies attempted to estimate the number of Panther tanks being produced . To do this they made use of the serial numbers printed on captured or destroyed German tanks . The principal numbers used were gearbox numbers , as these fell in two unbroken sequences . Chassis and engine numbers were also used , though their use was more complicated . Various other components were used to cross - check the analysis . Similar analyses were done on tires which were observed to be sequentially numbered ( i . e . , 1 , 2 , 3 , … , " N " ) . The analysis of Panther tank wheels yielded an estimate for the number of wheel molds that were in use . A discussion with British road wheel makers then estimated the number of wheels that could be produced from this many molds , which yielded the total number of Panthers that were being produced each month . Analysis of road wheels from the two tanks obtained ( consisting of 48 wheels each , for 96 wheels total ) yielded an estimate of 270 Panthers produced in February 1944 , substantially more than had previously been suspected . German records after the war showed production for the month of February 1944 was 276 . The statistical approach proved to be far more accurate than conventional intelligence methods , and the phrase German tank problem became accepted as a descriptor for this type of statistical analysis . Estimating production was not the only use of this serial number analysis . It was also used to understand German production more generally , including number of factories , relative importance of factories , length of supply chain ( based on lag between production and use ) , changes in production , and use of resources such as rubber . Specific data . According to conventional Allied intelligence estimates the Germans were producing around 1 , 400 tanks a month between June 1940 and September 1942 . Applying the formula below to the serial numbers of captured German tanks , ( both serviceable and destroyed ) the number was calculated to be 246 a month . After the war , captured German production figures from the ministry of Albert Speer showed the actual number to be 245 . Estimates for some specific months are given as : Similar analyses . Similar serial number analysis was used for other military equipment during World War II , most successfully for the V - 2 rocket . During World War II , German intelligence analyzed factory markings on Soviet military equipment , and during the Korean War , factory markings on Soviet equipment was again analyzed . The Soviets also estimated German tank production during World War II . In the 1980s , some Americans were given access to the production line of Israel ' s Merkava tanks . The production numbers were classified , but the tanks had serial numbers , allowing estimation of production . The formula has also been used in non - military contexts , for example to estimate the number of Commodore 64 computers ever built , where the result ( 12 . 5 million ) matches the official figures quite well . Countermeasures . To prevent serial number analysis , one can most simply not include serial numbers , or reduce auxiliary information that can be usable . Alternatively , one can design serial numbers that resist cryptanalysis , most effectively by randomly choosing numbers without replacement from a list that is much larger than the number of objects you produce ( compare the one - time pad ) , or simply produce random numbers and check them against the list of already assigned numbers ; collisions are likely to occur unless the number of digits possible is more than twice as many as the number of digits in the number of objects produced ( where the serial number can be in decimal , hexadecimal or indeed in any base ) ; see birthday problem . For this , one may use a cryptographically secure pseudorandom number generator . All these methods require a lookup table ( or breaking the cypher ) to back out from serial number to production order , which complicates use of serial numbers : one cannot simply recall a range of serial numbers , for instance , but instead must look them each up individually , or generate a list . Alternatively , one can simply use sequential serial numbers and encrypt them , which allows easy decoding , but then there is a known - plaintext attack : even if one starts from an arbitrary point , the plaintext has a pattern ( namely , numbers are in sequence ) . One example is given in Ken Follett ' s novel " Code to Zero " , where the encryption of the Jupiter C rocket serial numbers is described as : The code word here is Huntsville ( with repeated letters omitted ) to get a 10 - letter key . The rocket number 13 was therefore " HN " , or the rocket number 24 was " UT " . Frequentist analysis . Minimum - variance unbiased estimator . For point estimation ( estimating a single value for the total ) , the minimum - variance unbiased estimator ( MVUE , or UMVU estimator ) is given by : where " m " is the largest serial number observed ( sample maximum ) and " k " is the number of tanks observed ( sample size ) . Note that once a serial number has been observed , it is no longer in the pool and will not be observed again . This has a variance of so a standard deviation of approximately " N " / " k " , the ( population ) average size of a gap between samples ; compare " m " / " k " above . Intuition . The formula may be understood intuitively as : the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum , and written as This can be visualized by imagining that the samples are evenly spaced throughout the range , with additional samples just outside the range at 0 and " N " + 1 . If one starts with an initial gap being between 0 and the lowest sample ( sample minimum ) , then the average gap between samples is " m " / " k " − 1 ; the −1 being because one does not count the samples themselves in computing the gap between samples . This philosophy is formalized and generalized in the method of maximum spacing estimation . Derivation . The probability that the sample maximum equals " m " is formula _ 7 , where formula _ 8 is the binomial coefficient . The expected value of the sample maximum is Then is an unbiased estimator of " N " . To show that this is the UMVU estimator : Confidence intervals . Instead of , or in addition to , " point " estimation , one can do " interval " estimation , such as confidence interval s . These are easily computed , based on the observation that the odds that " k " samples will fall in an interval covering " p " of the range ( 0 ≤ " p " ≤ 1 ) is " p " ( assuming in this section that draws are " with " replacement , to simplify computations ; if draws are without replacement , this overstates the likelihood , and intervals will be overly conservative ) . Thus the sampling distribution of the quantile of the sample maximum is the graph " x " from 0 to 1 : the " p " th to " q " th quantile of the sample maximum " m " are the interval [ " p " " N " , " q " " N " ] . Inverting this yields the corresponding confidence interval for the population maximum of [ " m " / " q " , " m " / " p " ] . For example , taking the symmetric 95 % interval " p " = 2 . 5 % and " q " = 97 . 5 % for " k " = 5 yields formula _ 11 formula _ 12 , so a confidence interval of approximately formula _ 13 . The lower bound is very close to " m , " so more informative is the asymmetric confidence interval from " p " = 5 % to 100 % ; for " k " = 5 this yields formula _ 14 so the interval [ " m " , 1 . 82 " m " ] . More generally , the ( downward biased ) 95 % confidence interval is formula _ 15 . For a range of " k , " with the UMVU point estimator ( plus 1 for legibility ) for reference , this yields : Immediate observations are : Note that one " cannot " naively use " m " / " k " ( or rather ( " m " + " m " / " k " − 1 ) / " k " ) as an estimate of the standard error " SE " , as the standard error of an estimator is based on the " population " maximum ( a parameter ) , and using an estimate to estimate the error in that very estimate is circular reasoning . Bayesian analysis . The Bayesian approach to the German tank problem is to consider the probability formula _ 16 that the number of enemy tanks formula _ 17 is equal to formula _ 18 , when the number of observed tanks , formula _ 19 is equal to the number formula _ 20 , and the largest of the serial numbers formula _ 21 is equal to formula _ 22 . For brevity formula _ 16 is written formula _ 24 The rule for conditional probability gives The expression formula _ 26 is the conditional probability that the largest tank serial number observed is equal to formula _ 22 , when the number of enemy tanks is known to be equal to formula _ 18 , and formula _ 20 enemy tanks have been observed . It is where the binomial coefficient formula _ 31 is the number of formula _ 20 - sized samples from an formula _ 18 - sized population . The expression formula _ 34 is the probability that the maximum serial number is equal to " m " once " k " tanks have been observed but before the serial numbers have actually been observed . formula _ 35 can be removed from the formula by the following trick . The expression formula _ 37 is the probability that the total number of tanks is equal to " n " when " k " tanks have been observed but before the serial numbers have actually been observed . Assume that it is some discrete uniform distribution The upper limit formula _ 39 must be finite , because the function is which is not a probability mass function . Then If formula _ 43 , then the unwelcome variable formula _ 44 disappears from the expression . For " k " ≥ 1 the mode of the distribution of the number of enemy tanks is " m " . For " k " ≥ 2 , the probability that the number of enemy tanks is " equal to " formula _ 46 , is and the probability that the number of enemy tanks , formula _ 17 , is " greater than " formula _ 18 , is For " k " ≥ 3 , formula _ 51 has the finite mean value : For " k " ≥ 4 , formula _ 17 has the finite standard deviation : These formulas are derived below . Summation formula . The following important formula from Binomial coefficient # Identities involving binomial coefficients is used below for simplifying series relating to the German Tank Problem . This sum formula is somewhat analogous to the integral formula These formulas apply for " k " > 1 . One tank . If you observe one tank randomly out of a population of " n " tanks , you get serial number " m " with probability 1 / " n " for " m " ≤ " n " , and zero probability for " m " > " n " . Using Iverson bracket notation this is written This is the conditional probability mass function of formula _ 22 . When considered a function of " n " for fixed " m " this is a likelihood function . The maximum likelihood estimate for the total number of tanks is " N " = " m " . The total likelihood is infinite , being a tail of the harmonic series . but where formula _ 62 is the harmonic number . The probability mass function depends explicitly on the prior limit formula _ 44 : The mean value of formula _ 17 is Two tanks . If you observe two tanks rather than one , then the probability that the larger of the observed two serial numbers is equal to " m " , is When considered a function of " n " for fixed " m " this is a likelihood function The total likelihood is and the probability mass function is The median formula _ 71 satisfies so and so the median is but the mean value of " N " is infinite Many tanks . Probability mass function . The conditional probability that the largest of " k " observations taken from the serial numbers { 1 , … , " n " } , is equal to " m " , is The likelihood function of " n " is the same expression The total likelihood is finite for " k " ≥ 2 : The probability mass function is The complementary cumulative distribution function is the probability that " N " > " x " The cumulative distribution function is the probability that " N " ≤ " x " Expected value and standard deviation . The expected number of enemy tanks is The standard deviation σ satisfies the equation So and the standard deviation is The variance - to - mean ratio is simply