sorry for being such a useless blogger lately. I started off posting as often as a few times a week and now it’s been over a month since my last post! It could be the three distinct talks that I’ve had to give recently, or that my PhD is reaching crunch time, or the fact that I now have a girlfriend; somehow I’ve not found the time to blog though. As I’m currently visiting a GR program at MSRI and staying in a house in Berkeley Hills, away from distractions, I can type up a post I had sitting in the ideas pile.

Pick a number – any number…

What did you pick? Was it 7? Or pi? or $1+e^{i\pi}$? I mean, pick ANY number. It doesn’t have to be a whole number, or even a rational number. If you were to choose a random number — a truly random number — out of all of the real numbers, then the probability that it was a whole number, or even a rational number, is zero. Take a look at My Infinity is Bigger Than Yours to figure that one out, because that’s not exactly the problem I’ll be talking about today. What I’m going to convince you of today, is that the number will certainly contain a 7 as a digit — 100%.

Let me explain.
The random number can – and will be – arbitrarily large so it will have many many digits. If I had a 100 digit number then the probability that the first digit is not a 7 is 9/10, the same for the second digit and the same for the rest. So the probability that none of the digits are 7 is given by the product of the individual probabilities — $(9/10)^{100}=0.00003$. That’s a 3 in 100,000 chance. And if we start looking at numbers with 10000000000000 digits, then this probability quickly goes to zero.

Alternatively, you could consider how many numbers less than 10 have a 7; then how many less than 100 have a 7, etc. and you would see that this ratio approaches 1 as you start including higher and higher numbers.

Then again, does it even make sense to “pick a random number”?