A sports riddle

Here’s one of my favorite thinking riddles of all time. I like it so much I have probably written about it before in internet forums or even in this blog. I haven’t posted anything in the blog in a while. I haven’t played any poker, and only made a few small bets in the NCAA tournament. The baby is keeping my wife and I very busy! Here’s the riddle.

Two teams play in a best-of-seven series in which each team has exactly 50% chance of winning each game, no matter what happened in the previous games. What is more likely: the series ends in 6 games or the series ends in 7 games?

Try to think about the question and answer it without doing any math. Give yourself at least a minute to think about it before reading the answer below.

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For most people, the intuitive, gut-feel first answer is that it is that the series is more likely to end in 7 games than 6 games. But the correct answer is that it is equally likely. One can use brute force and crank out all the possibilities and come up with that answer. But here is a more elegant answer that has little math involved.

How can a series end in 7 games? The answer is that the series must be tied 3-3 in order for the 7th game to be played. How can a series get to 3-3? It can only get to 3-3 when one team is ahead 3-2. When one team is ahead 3-2, then if they win that 6th game, they win the series. If the lose that 6th game, then the series ends in 7 games. Since each team has a 50% chance of winning each game, that means there is a 50% chance for the team ahead 3-2 in the series to win and end it in 6 games, and a 50% chance for the team ahead 3-2 in the series to lose and have a 7th game.