The cereal box problem: How many boxes does it take to find all prizes?

Children’s cereal manufacturers often attract the attention of young clients by including small prizes and toys in every box; sometimes all prizes are identical, but most often individual prizes are part of a collection, and kids are encouraged to collect them and try to complete a full collection. How long does it take ?

Simple probabilistic modeling shows that on average $$n (1 + 1/2 + \ldots + 1/n)$$ boxes are required to complete a full set of $$n$$ prizes: for example, it takes on average $$14.7$$ boxes to complete a full set of six prizes.

An experimental estimation of the entropy of English, in 50 lines of Python code

“Th_ onl_ wa_ to ge_ ri_ of a tempta____ is to yie__ to it. Resi__ it, an_ you_ soul gro__ sic_ wi__ longi__ fo_ th_ thin__ it ha_ forbi____ to itse__.”

(Osc__ Wil__, The Picture __ ______ ____)

Thanks to the verbosity of the English language, proficient English speakers generally find it relatively easy to decipher the above passage despite the numerous omissions.

How does one quantify this redundancy? This article introduces the notions of Shannon entropy and information rate, and experimentally estimates the information rate of written English by training a Markov model on a large corpus of English texts. This model is finally used to generate gibberish that presents all the statistical properties of written English. Best of all, the entire source code fits in 50 lines of elegant Python code.

Modeling and measuring string comparison performance in C, C++, C# and Python.

Comparing strings is often — erroneously — said to be a costly process. In this article I derive the theoretical asymptotic cost of comparing random strings of arbitrary length, and measure it in C, C++, C# and Python.

Generating uniformly random data from skewed input: biased coins, loaded dice, skew correction, and the Von Neumann extractor

In a famous article published 1951, John Von Neumann presented a way of skew-correcting a stream of random digits so as to ensure that 0s and 1s appeared with equal probability. This article introduces a simple and mentally workable generalization of his technique to random dice, so a loaded die can be used to uniformly draw numbers from the set $$\{1, 2, 3, 4, 5, 6\}$$, with reasonable success.