The Golden State Warriors lost to the Cleveland Cavaliers in basketball last night. They're now down 2-1 in the series. The first team to win 4 is the champion.
I don't care about basketball but there's one thing I like about the game and that's that scores generally increase relatively at random (I hope -- I hope the near miraculous comeback yesterday from a 17-point deficit after three quarters wasn't programmed), and that games are won to some degree seemingly at random. I like random.
As an aside, I do find it remarkable in basketball how often a team with a big deficit claws its way back only to lose in the end by a small margin. I'd like to see a statistical analysis of this. A huge amount of money is at stake for games not being a total blow-out. I do wonder at this. Basketball has long seemed to me to be more about the show and less about a fair contest. And that makes it very difficult for me to care about the result. But this is an aside.
Assuming games are fair, what's the chance of Golden State winning the series?
Assume each team has an equal chance to win games. You can weight results under different assumptions but this is simplest.
Then the key to solving this problem is to recognize that while the series is terminated when the first team reaches 4 wins, this is irrelevant to the odds of who wins. It's easier to calculate the odds of winning if you assume the series always goes 7 games. If you assume that, it becomes a simpler probability problem.
Then all sequences of win-loss are equally probably. Each has a chance 2-n, where n is the number of remaining aims. And since 3 games have been played, 4 remain (I'm assuming we always play the full 7). So I need to count how many of those possible 4-game sequences result in the Warriors winning.
There's one sequence where the warriors win all 4. They win the series then, 5-2.
There's four sequences where the warriors win 3. Then Cleveland wins 1. That win can occur in either game 4, 5, 6, or 7.
The rest of the possible sequences the Warriors fail to win at least three games. So they win the series in 5 of the 16 options. Since each is equally likely to occur, they win in 31.25% of the possible results. That's their chance to win the overall, assuming games are completely random.
To calculate it assumingt he series ends as soon as one team gets to 4 is more complicated. You'd need to start with the odds the warriors win the next 3 games. Then consider the odds they win exactly 2 of the next 3 but then win the 4th. Then add these together.
I can do this: the odds of winning 3 in a row is 1/8 = 2/16. The odds of winning exactly 2/3 is 3/8 (there's 3 ways to lose one game of 3, and there's 8 ways total the 3 games could end up, so that's 3/8). Then I need to divide by 2 because they then need to win the 7th game, which has 50% chances. So that's net 3/16. So the total chances of winning are 2/16 + 3/16 = 5/16. This is the same answer as I got before but is more convoluted.
The principle that cutting the series short can have no effect on the winner, assuming results are random with a fixed chance for each team to win, has more implications than just simplicity in calculating the odds. It also means the sequence of home versus away games doesn't matter. There's been a lot of debate in various sports whether a 2-3-2 sequence of teams A and B being home, which generally has an advantage, is more balanced than a sequence 2-2-2-1. After all, 2-3-2 could result in team B having 3 home games to team A's 2 home games if the series ends after 5. However, if I view it as the series always goes 7 games then it seems the sequence should not matter. And since going to a full 7 never changes the result I therefore conclude the sequence doesn't matter. You can just as well go 3-4. The home field advantage for the series is alwsys with the team with 4.