Parrondo’s Paradox : A combination of losing strategies becomes a winning strategy. This paradox can be seen via a simple simulation of three games. Game Type 1 – You flip a biased coin that has 1/2-epsilon as the probability of heads. For each head you get $1 and for each tail you lose $1 Game Type 2 – If your capital is a multiple of 3, you flip a biased coin that has 1/10-epsilon, as probability of heads.

Computing steady state probabilities for a queueing system is somewhat easier than computing the transient distributions. The latter typically comprises a differential-difference equation and the usual trick of recursive substitution fails as there is a derivative in the equation. The tools employed in solving a differential-difference equation are Generating functions, Laplace transforms and PDE solving tricks. Only for simple systems such as M/M/1 can one slog out and find a closed form solution.

[youtube https://www.youtube.com/watch?v=B0hMS4eUVV8?rel=0]

In a book that has about 350 pages, the first 250 odd pages are devoted to probability, ODEs and difference equations. The last part of the book covers queuing theory for specific systems, i.e, Poisson arrivals, exponential service times of one or more servers. The most painful thing about this book is that there are innumerable typos. A book that is riddled with typos on almost every other page cannot be an appealing text for an undergrad.

This puzzle has appeared in many forms but here is one variant: Five sailors were cast away on an island. To provide food, they collected all the coconuts they could find. During the night one of the sailors awoke and decided to take his share of the coconuts. He divided the nuts into five equal piles and discovered that one nut was left over, so he threw this extra one to the monkeys.