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

In a famous article published 1951 ((Various techniques used in connection with random digits. NIST journal, Applied Math Series, 12:36-38, 1951. This article does not seem to be available online, though it is widely cited. It is reprinted in pages 768-770 of Von Neumann’s collected works, Vol. 5, Pergamon Press 1961)), 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.

# The cafeteria paradox: stop using the water dispenser while someone else does!

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?