I had to dig a little deeper to see what Dave meant about linearity of expectation, but when I did I found an incredible solution to the problem in just a couple of tweets:

Prior to this series of tweets from Dave, probably the most interesting “linearity of expectation” example that I’d seen was a new-to-me proof of the Buffon Needle problem in Jordan Ellenberg’s How not to be Wrong.

Although I don’t have the exact reference in Ellenberg’s book (I have the audio book version), Lior Pachter has more or less the same neat explanation on his blog. It is amazing to me that changing the needle to a circle solves the problem in a snap!

Anyway, a couple of points about problem in Dave’s tweets. First, it wouldn’t have occurred to me to use linearity of expectation to attack this problem, and that is definitely my bad. Dave’s example really showed me the power of the approach. Second, my intuition for an approximate answer to the question was off by miles! It actually took a while for me to understand how the number could be as high as 488 (I mean, you’ve got a 50% chance to win after about 500,000 tries, and, really, how many different people could have won 10 times by then . . . .), but I’m glad that Dave’s tweet made me think about this problem – I definitely needed the intuition adjustment!