## Searching $latex \pi$

Take a look at the Pi Search Page. You can type in a bunch of digits and it tries to find them somewhere in the first 3.2 billion digits of the decimal expansion of $\pi$. $\pi$, of course, is the ratio of any circle’s circumference to its diameter, and is approximately equal to $\pi \approx 3.14159265358979323846264\dots$

The digits never repeat since $\pi$ is irrational. Try searching for your 7-digit phone number. Surprised that it was found? Even more surprised that I was so sure your phone number would be found?

Well, let’s think about it for a minute. Assuming the digits of $\pi$ are essentially random, the chance of a particular digit having a particular value is 1 in 10; in other words, the probability of a particular digit having a particular value is $1/10$. So the probability that 7 digits in a row will all have certain values (such as your phone number) is $(1/10)^7$, or one chance in ten million. This might sound like a very small chance, but remember, we are talking about 3.2 billion digits in which to search. So in fact, we would expect that your phone number occurs about $3.2 \times 10^9 / 10^7 = 320$ times in the first 3.2 billion digits of $\pi$! This also means that you have a good chance of finding any eight-digit sequence of numbers, and a fairly good chance of finding any nine-digit sequence of numbers, but your chances of finding any particular sequence of ten or more digits are not so good. Try it and see! 