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.
“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.
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.
Most cafeteria water dispensers will let two (sometimes more) people fill a jug at the same time. This article uses simple maths to prove that it’s a waste of time. In other words, two people should never use the same water dispenser at the same time: I call this the cafeteria paradox.
An intuitive presentation
Let’s start with a little brain teaser:
Alice and Bob are at the cafeteria, seating at different tables. Alice stands up to refill her table’s water jug. Little after, Bob stands up with his own table’s jug, heading to the same water dispenser. That water dispenser is a perfectly standard one, with two taps, and Bob finds himself standing near Alice. After a small hesitation, Bob starts using the second tap to fill his own jug, thereby diverting part of the output previously devoted to Alice.
This grants him an exasperated and somewhat puzzled look from Alice. Why?