## Sunday, April 25, 2021

### 269: A Good Use of Downtime

The recent observation of Holocaust Remembrance Day reminded me of one of the stranger stories to come out of World War II.    John Kerrich was a South African mathematician who made the ill-fated decision to visit some family members in Denmark in 1940 just before the Germans invaded, and soon found himself interned in a prison camp.    We should point out that he was one of the luckier ones, as the Germans allowed the Danes to run their prison camps locally, and thus lived in extremely humane conditions compared to the majority of prisoners in German-occupied territories,   But being imprisoned still left him with many hours of time to fill over the course of the war.     Kerrich decided to fill this time by doing some experiments to demonstrate the laws of probability.

Kerrich’s main experiment was a very simple one:   he and a fellow prisoner, Eric Chirstensen, flipped a coin ten thousand times and recorded the results.    Now you might scratch your head in confusion when first hearing this— why would someone bother with such an experiment, when it’s so easy for anyone to do at home?    We need to keep in mind that back in 1940, the idea that everyone would have a computer at home (or, as we now do, in their pocket) that they could use for seemingly endless numbers of simulated coin flips, would have seemed like a crazy sci-fi fantasy.    Back then, most people had to manually engage in a physical coin flip or roll a die to generate a random number, a very tedious process.      Technically there were some advanced computers under development at the time that could do the simulation if programmed, but these were being run under highly classified conditions by major government entities.   So recording the value of ten thousand coin flips actually did seem like a useful contribution to math and science at the time.

So, what did Kerrich accomplish with his coin flips?    The main purpose was to demonstrate the Law of Large Numbers.  This is the theorem that says that if you perform an experiment a large number of times, the average result will asymptotically approach the expected value.    In other words, if you have a coin that has 50-50 odds of coming up heads or tails, if you perform lots of trials, you will over time get closer and closer to 50% heads and 50% tails.   Kerrich’s coins got precisely 5,067 heads, and over the course of the experiment got closer and closer to the 50-50 ratio, thus providing reasonable evidence for the Law.    (In any 10000 flips, there is about an 18% chance that we will be off by at least this amount from the precise 50-50 ratio, so this result is reasonable for a single trial.)

Of course, it might make sense to request another trial of 10000 flips to confirm, for improved confidence in the result.   But apparently even in prison you don’t get bored enough for that— in his book, Kerrich wrote, “A way of answering the… question would be for the original experimenter to obtain a second sequence of 10000 spins,   Now it takes a long time to spin a coin 10000 times, and the man who did it objects strenuously to having to take the trouble of preparing further sets.”

Kerrich and Christensen also did some other experiments along similar lines.    By constructing a fake coin with wood and lead, they created a biased coin to flip, and over the course of many flips demonstrated 70/30 odds for the two sides.   This experiment was probably less interesting because, unlike a standard coin, there likely wasn’t a good way to estimate its expected probabilities before the flips.   A more interesting experiment was the demonstration of Bayes’ Theorem using colored ping-pong balls in a box.   This theorem, as you may recall, helps us calculate the probability of an event when you have some knowledge prior conditions that affect the likelihood of each outcome.    The simple coin flip experiment seems to be the one that has resonated the most with reporters on the Internet though, perhaps because it’s the easiest to understand for anyone without much math background.

In 1946, after the end of the war, Kerrich published his book, “An Experimental Introduction to the Theory of Probability.”.    Again, while it may seem silly these days to worry about publishing experimental confirmation of something so easy to simulate, and which has been theoretically proven on paper with very high confidence anyway, this really did seem like a useful contribution in the days before widespread computers.    The book seems intended for college math students seeking an introduction to probability, and in it Kerrich goes over many basics of the field as demonstrated by his simple experiments.    If you’re curious about the details and the graphs of Kerrich’s results, you can read the book online at openlibrary.org, or click the link in our show notes at mathmutation.com .   Overall, we have to give credit to Kerrich for managing to do something mathematically useful during his World War II imprisonment.

And this has been your math mutation for today.

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