It is usually taught in statistics classes that Binomial probabilities can be approximated by Poisson probabilities, which are generally easier to calculate. This approximation is valid “when (n) is large and (np) is small,” and rules of thumb are sometimes given.
In this post I’ll walk through a simple proof showing that the Poisson distribution is really just the Binomial with (n) (the number of trials) approaching infinity and (p) (the probability of success in each trail) approaching zero.
I present a solution to a modification of the “hardest logic puzzle ever” using probability theory.
Background “The hardest logic puzzle” was originally presented by Boolos (1996) and since then it has been amended several times in order to make it harder (see B. Rabern and Rabern 2008, Novozhilov (2012)).
The puzzle: Three gods A, B, and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter.