## Suggested Reading: One Hundred Essential Things You Didn’t Know you Didn’t Know: Maths Explains Your World

John D. Barrow — One Hundred Essential Things You Didn’t Know you Didn’t Know: Maths Explains Your World — Norton, 2008, 284 pp. ISBN 978-0-393-07007-1

1 One I like particularly is the trick proposed by von Neumann to transform a biased coin into a fair coin. Let’s say we have a coin that lands on head with probability $p$ and on tail with probability $1-p$ (both quite far from $\frac{1}{2}$). Von Neumann observes that if you throw the coin twice, it will land twice on heads with probability $p^2$, twice on tails with probability $(1-p)^2$. But head followed by tail and tail followed by head have the same probability of $p(1-p)$. The quantity $p(1-p)$ isn’t $\frac{1}{2}$, but if each time we throw two consecutive heads or tails we “forget” them, keeping only the draws with one head and one tail (HT or TH), we get an unbiased, or fair, coin. Suffice to map HT to heads and TH to tails (or vice-versa) and we’re done.