Hello, mgoblog. I've always wondered about these statistics, which we hear about this time of year re: NBA and NHL playoff series, which say something like the team that wins game one (in the NBA) wins the series 78% of the time -- this is actually something I've heard tonight. Of course, the team that wins any particular game wins the series most of the time, so without knowing what % we would expect for each game purely by chance, we can't assess the significance of something like this.
Tonight, I decided to solve this problem. It's very easy to do, and maybe others have done it, but I thought I'd share it here, because I'm a U-M grad, and I'd guess someone here will find it interesting. Cheers.
I wrote a simple code to simulate best-of-seven series in which the result of each game was pure chance -- a 50% coinflip. I don't know how to analytically solve this problem, but luckily in the modern age that's not an important skill. You just have to find a way to brute force it stupidly with a computer. So I simulated 100 million best-of-7 series in this way, in which the result of each game was totally random. Then I asked what is the probability of winning the series, given that you have won game X?
I don't know why this is the answer, but this is the answer:
Game 1: 65.6%
Game 2: 65.6%
Game 3: 65.6%
Game 4: 65.6%
Game 5: 67.85%
Game 6: 75.00%
Game 7: 100%
I'm not sure why these numbers look like they do; I'd be interested if anyone here feels like wading into the theory. But it seems like, for the first few games of a best-of-seven series, pure chance suggests that winning a game predicts a 65% chance of winning the series. Anything significantly above that would suggest that a particular game -- like game one in the NBA -- carries unusual importance. Of course, this is only a preliminary, superficial analysis that does not take into account important factors such as home advantage. But it is an important first step which I have never seen anyone actually take.