The German Tank Problem and Kenyan PSVs

By Chris Orwa
  Published 18 Feb 2016
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A great story unfolds at the heart of World War II. Following Operation Barbarossa (the invasion of Soviet Union by Germany), Joseph Stalin ordered a new type of tank to counter the German blitzkrieg (lightning war) – an innovative fighting method of concentrated firepower coupled with highly mobile tanks. This worked effectively in conquering the French by gaining the element of surprise.  However, when Germans attacked the City of Kursk, they met the T-34 Soviet medium tank that was faster, had thicker armor and packed a big punch.

Adolf Hitler was surprised! He gathered automotive manufacturers in Germany and ordered for a new tank design capable of obliterating the T-34 tank. Porsche Industries, Henschel & Sohn, and Krupp Industries teamed up to build the now infamous Tiger I tank (panzerkampfwagen). Its high-velocity gun and impenetrable armor made it invincible in battle. The deafening bang from its shell also spread a new disease in the battlefield – tigerphobia.

Allied forces did not possess a tank that could take on the Tiger. However, the Americans soon discovered a weakness in the Tiger design – the rear engine compartment was not armored. The generals quickly developed a Tiger hunting plan. It entailed using 3 tanks as bait and a fourth tank would sneak to the rear and blow-up the Tiger. Several successful hunts led to widespread adaptation of the plan. Since the Americans lost 3 tanks for every Tiger destroyed, they needed to know how many were being produced.

The task fell to Richard Ruggles, a Harvard economist working at the Economic Warfare Division in London. His initial task involved confirming reports from espionage activities in German factories and confessions from prisoners of war. These data capture methods did not seem to provide reliable and consistent projections. Richard then proceeded to estimate production using pre-war records on factory capacities – it was a dead end.

Frustration in predicting the number of Tiger tanks produced every month led to the coining of the term The German Tank Problem – a mathematical puzzle on how to estimate the total size of a given population from partial observation. Richard cracked the puzzle in 1943. He asked field soldiers to send photographs of captured tanks, from whence he used the serial numbers on the tanks to create a simple yet effective formula for estimating monthly productions.

Formula for Solving the German Tank Problem Formula for Solving the German Tank Problem

He later wrote a paper with Henry Brodie from the Department of State titled ‘An Empirical Approach to Economic Intelligence in World War II’ that discusses the details of the analysis. The formula seeks to evaluate if the largest observed number is the highest in the production series. In statistics, this is method is referred to as the maximum likelihood estimation – a process of estimating population parameters based on observed data from its probability distribution. The tank estimation formula may be understood intuitively as the sample maximum plus the average gap between observations in the sample.

This simple formula greatly contributed to the Allied war efforts in subduing the Germans. After the war, production records captured from the Albert Speer ministry confirmed the accuracy of the formula. Actual records indicated 255 tanks were built every month whilst the formula predicted 256. It is worthnoting that conventional intelligence had placed German production at 1500 every month.

Kenyan Public Service Vehicles
While walking home on a jam-packed evening, I realized that Kenyan PSV buses bore a fleet number – quite similar to the German tanks. I figured I could estimate the number of buses each bus company had and study who’s dominating the public transport market. Armed with a pen, notebook and camera, I traversed the busy Nairobi city streets recording plate numbers and fleet number from bus companies.


Using the estimation formula, I estimated the fleet number of bus companies in Nairobi as shown in the graph below. KBS has the largest number of fleet buses in the city closely followed by City Shuttle. Amazingly Citi Hoppa comes at third yet it was the dominant public transport company in Nairobi in the recent past.

Fleet Number of Each Bus Company Fleet Number of Each Bus Company

The Competition
To better understand the competition, it would  be prudent to measure the rate at which each bus company updates their fleet. The graphs below mirrors the ‘growth rate’ of the company fleet.

Kenya Bus Service Kenya Bus Service
Citi Hoppa Citi Hoppa
Forward Travelers Forward Travelers
City Shuttle City Shuttle

The Kenya Bus Company (KBS) has a steady growth on its fleet hinting at constant investment in acquiring new buses. As alluded to earlier, Citi Hoppa had an almost flattening growth rate until sometime last year – perhaps due to injection of fresh capital. Forward Traveler than mostly ply Kayole and Eastleigh routes had an initial high growth rate of buses but has since leveled off – a good guess would be competition from newer vehicles from Ummoiner sacco. Lastly, Citi Shuttle has the newest set of buses and a good growth rate. So who wins in this battle of covering the city? I’ll wager KBS.

Nairobi has 24 registered bus companies/saccos, this analysis was limited to buses with fleet numbers affixed to the exterior. The following bus companies have a considerable large fleet but don’t affix fleet number on their bus exterior.

  • Ummoiner
  • Embassava
  • Eastern Bypass
  • NMOA
  • Double M

Credit: Henk Harmsen
Photo by: Centre of Math Blog
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