Deceptive numbers (2)

Another example from the courtroom. Somewhere in the US outback a local railway company was taken to court over their plans to increase the number of trains on the local line. The regulations stated that increasing train traffic “by less than 100%” could be done without consulting the local community. In other words unless the number of trains at least doubled consultation was not required. The problem was that the company proposed to send two trains a day down an unused line.

Having duly considered the matter the judge argued that to determine the ratio of train traffic increase one would need to divide the proposed number of trains by the current traffic. In the case at hand the equation presents as 2/0 which, as we know, has no numeric result because it is impossible to divide by zero. In absence of a meaningful mathematical formulation the judge opined that it was impossible to determine if the proposed increase exceeded the 100% threshold and so no consultation was required. This is one way of looking at the problem but let’s turn things around. What number of trains a day constitutes a 100% increase on the original traffic? 2*0=0, so zero trains a day is the threshold triggering the local consultation requirement. Two trains are more than zero trains so, following this logic, the increase in train traffic is more than 100%. The judge got it wrong.

And now a funny one. Let’s imagine a medical condition which inevitably leads to a sudden (and messy) death. It is known as Head Exploding Syndrome or, in short – HES. Luckily it is incredibly rare, affecting only 1-in-one billion humans. There is a test which can diagnose HES with a very low error rate of 1-in-one million. You happen to have been diagnosed with HES using the test. Things look ominous but what is the actual probability of your head blowing up? The condition is so rare that we need to consider false positives. If the world population were 6 billion and all humans were tested we could expect 6 000 000 000 / 1 000 000 000 = 6 “true” positives. However, one-in-a million of those who do not have HES would get a “false” positive result – this is 5 999 999 994 / 1 000 000 = 6000 tests. In all, 6 + 6000 = 6006 people would test positive but only 6 of them would actually have HES – a 6 / 6006 = 0.1% probability. Not as bad as it first looked!



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