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I have a lot of coin-tossing problems floating around in my head, of varying levels of difficulty. Here’s one which requires some thought, but shouldn’t present too much of a challenge if you’ve seen this kind of problem before.

You start flipping a fair coin, and keep going until you’ve seen n heads in a row, when you stop. On average, how many times do you flip the coin before stopping?

Spaghetti

This is what spaghetti looks like

Here’s a nice problem in probability. I like it because at first sight it seems fiercely complicated, but thinking about it in the right way makes it seem much simpler.

You have a bowl filled with n strands of spaghetti. You pick two ends from the bowl at random, and tie them together. You keep doing this until there are no more loose ends. Obviously, you will have made some number of loops when you reach the end of this process.

What is the average number of loops created?