You may have heard of the "infinite monkey theorem". It states: if you take a monkey, give it a typewriter and an infinite amount of time, sooner or later it will type out the complete text of Hamlet.^[1]

It's said that this theorem emerged during 19th-century debates about evolution. Proponents of evolution supposedly argued that human descent from apes, like other low-probability events, is possible – you just need to allow enough time.

I began my talk about low-probability events with this example at an informal seminar for graduate students at UCLA. I'll share this talk with you as well. My story will have several parts; the first part is dedicated to Buridan's principle.

A safety-critical system is one whose failure could result in death or serious harm to human health. Examples include nuclear power plants, airplanes, railway crossings, and pacemakers. In the aviation industry, the guidelines want death to be an extremely improbable failure condition and those are defined as having a probability on the order of $10^{- 9}$ or less. So, a safety-critical system should be designed to cause no more than one death per billion hours of operation. Engineers work hard at this, calculating all these nanomorts per hour. When reading about this, a natural question arises – why can't the risk simply be reduced to zero? Is there really such a significant gap between $0$ and $0.000000001$ ?

Buridan's ass

It turns out this gap exists, and it's substantial. The problem is that the laws of classical physics are continuous. As Feynman said in the introduction to his famous lecture series: "All things are made of atoms – little particles that are in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another". The interactions between these atoms can be modeled with various laws. Atoms experience forces and move according to Newton's laws. They interact with the electromagnetic field, which also interacts with itself according to Maxwell's equations. All these laws share one property – they are continuous. If we continuously move some parameter, such as the initial position of an atom, its position after 10 seconds will depend on this parameter continuously.

From this simple observation follows the paradox of Buridan's ass. The French philosopher Jean Buridan reportedly claimed that if a hungry ...

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