Image credit: “Coins,” by Jason Rogers, on flickr.

You’re sitting on the subway. Your phone has low battery, and you have nothing to read. How do you kill time?

I like to carry around a puzzle with me. Not a physical puzzle, like a jigsaw or a Rubik’s cube. A mental puzzle– something to think about for fun. The nice thing about that kind of puzzle is that it’s easy to bring anywhere. I can think about it while riding the train, taking a walk, or sitting in the dentist’s chair.

Here is the type of puzzle I’m talking about. It’s from the 2010 CUNY math contest. This particular puzzle has a little story involved.

Alice and Bob are bored and want to play a game. “I have an idea,” Alice chimes in. “How about you flip this coin 2009 times and I’ll flip it 2010 times and whoever gets more Heads wins?”
Bob replies, “No, that’s not fair! You’re probably going to win since you get more flips!”
“Fine!” answers Alice. “How about this? You flip the coin 2009 times and I’ll flip it 2010 times and if I get more Heads, I win. If you get more Heads, you win. And, if there’s a tie, we’ll say that you win, too.” Bob shrugs his shoulders and agrees to play.
What’s the probability that Alice wins the game? Prove your answer.

The one thing that might seem unfamiliar: “Prove your answer.” In this situation, that means give a reason for the probability in your answer. For example, if I think the odds of Alice winning are .75, I should be able to logically explain why that is so, in a way that rules out all other possibilities.

Feel free to ask questions in the comments! I’ll share a solution in a later post.