With the Stanley Cup just underway, the NBA Finals about to begin, and the author still bitter over the 2017 World Series, what better occasion than this to write about the playoff format that each of these competitions uses to determine its champion: the seven-game series.

The rules of the seven-game series are simple: two teams play each other until one of the teams wins four games. As ties are not permitted, it will take at least four games and at most seven games to determine a champion. (The second part of this statement is not completely trivial and worth at least a moment’s thought.)

Assuming both teams are equally matched (even though this is almost never true!) and that home field/ice/court advantage plays no role (again, almost never true), each team enters the series with a 50% chance of winning. Once a series is underway (e.g., the Las Vegas Knights just took a 1-0 series lead last night), a good question to ask is how this probability changes. For example, now that the Knights lead the series 1-0, what is the probability they will win the series?

Before examining that particular question, we’ll start with a much easier one and then build on our method to solve harder problems.

The 3-2 series lead

When a team leads the series 3-2, they will win the series unless they drop the next two games in a row. Assuming the probability of dropping each game is 1/2, then the probability of dropping both games is 1/2 x 1/2 = 1/4. Therefore, the probability of winning–i.e., the probability that the team does NOT drop the next two games–is 1 – 1/4 = 3/4.

However, rather that use the above approach to solve the problem, we will take a different approach that generalizes more easily.

There are up to two more games remaining in the series since five have already been played. Forgetting for a second that the first two of these outcomes could not actually arise within the rules of a seven-game series (why?), the four equally probable outcomes when two games are played are–

W-W  W-L  L-W  L-L

 

 

 

Because the first three of these will result in four or more wins for the team leading 3-2, the probability of winning the series is 3/4. Conversely, it is easy to see that the probability of winning the series for the team trailing 3-2 is 1/4.

The 3-1 series lead

Image result for lebron 2016 nba champMuch was made of the 3-1 leads the NBA’s Oklahoma City Thunder  and Golden State Warriors blew in the 2016 Western Conference Finals and NBA Finals respectively. And sure enough, losing the series from a 3-1 advantage should be a fairly rare event (though not as rare as it’s made out to be).

A shortcut to finding the probability of winning is again to recognize it as the complement of losing three games in a row:

  • Probability of L-L-L = 1/2 x 1/2 x 1/2 = 1/8
  • Probability of not L-L-L = 1 – 1/8 = 7/8.

However, we will again take a slower path to this same result. Having played four games, there are now up to three games remaining. All possible outcomes–valid or not in this context–for three games are–

W-W-W   W-W-L  W-L-W  W-L-L  L-W-W  L-W-L  L-L-W  L-L-L

As before, all but the last of these eight outcomes results in the team with the 3-1 lead finishing with at least four wins. Therefore, the probability of winning the series is 7/8.

The 1-0 series lead

Though the overall method is unchanged, the most time-consuming problem is to compute the probability of winning the series following a 1-0 lead. This is because there are up to six games remaining, hence 2^6 = 64 possible outcomes to consider. Though there are (always!) clever shortcuts available, it will work here to list all 64. Entries in red result in at least four wins for the team with the 1-0 lead.

W-W-W-W-W-W   W-W-W-W-W-L   W-W-W-W-L-W   W-W-W-W-L-L   
W-W-W-L-W-W   W-W-W-L-W-L   W-W-W-L-L-W   W-W-W-L-L-L  
W-W-L-W-W-W   W-W-L-W-W-L   W-W-L-W-L-W   W-W-L-W-L-L  
W-W-L-L-W-W   W-W-L-L-W-L   W-W-L-L-L-W   W-W-L-L-L-L 
W-L-W-W-W-W   W-L-W-W-W-L   W-L-W-W-L-W   W-L-W-W-L-L   
W-L-W-L-W-W   W-L-W-L-W-L   W-L-W-L-L-W   W-L-W-L-L-L 
W-L-L-W-W-W   W-L-L-W-W-L   W-L-L-W-L-W   W-L-L-W-L-L 
W-L-L-L-W-W   W-L-L-L-W-L   W-L-L-L-L-W   W-L-L-L-L-L 
L-W-W-W-W-W   L-W-W-W-W-L   L-W-W-W-L-W   L-W-W-W-L-L   
L-W-W-L-W-W   L-W-W-L-W-L   L-W-W-L-L-W   L-W-W-L-L-L 
L-W-L-W-W-W   L-W-L-W-W-L   L-W-L-W-L-W   L-W-L-W-L-L 
L-W-L-L-W-W   L-W-L-L-W-L   L-W-L-L-L-W   L-W-L-L-L-L 
L-L-W-W-W-W   L-L-W-W-W-L   L-L-W-W-L-W   L-L-W-W-L-L   
L-L-W-L-W-W   L-L-W-L-W-L   L-L-W-L-L-W   L-L-W-L-L-L 
L-L-L-W-W-W   L-L-L-W-W-L   L-L-L-W-L-W   L-L-L-W-L-L 
L-L-L-L-W-W   L-L-L-L-W-L   L-L-L-L-L-W   L-L-L-L-L-L

The 45 entries in red correspond to a 42/64 (about 66%) probability of winning the series.

Results

The table below summarizes the results from above and inserts probabilities for cases not specifically addressed. All values are rounded to the nearest whole percent. As an example of how to read the table, if your team is up 2-0, you would look at the intersection of the “2” column for your team and the “0” row for the other team. The value of 81% would correspond to your team’s probability of winning the series. To find the other team’s probability of winning the series, you can simply subtract your team’s result from 100%. (Equally, one could look at the 0-2 entry in the table.)

Capture

Pivotal games

Occasionally you will hear things about a particular game in the series being the most pivotal or most important. Of course, the single most important game–when it happens–is Game 7 since it takes either team from a 50% probability to either 100% or 0%, a shift of 50% for each team. The impact of other games in a series depends on the state of the series to that point. We can examine the impact by using the same table from above and recognizing that the result of any particular game corresponds to moving one entry to the right or down. For example, after your team takes a 2-1 lead, a win in Game 3 (green arrow) increases the probability by 19% and a loss decreases it by this same amount (red arrow).

Capture

In general, the deeper teams are in the series, the more any particular game will matter. However, this is more a general trend than an absolute. We have already identified Game 7 as the highest impact game. Here are the impacts of the other games on the teams that win them–

  • Game 1 – 16%
  • Game 2 – 15% for a team up 1-0; 16% for a team down 0-1.
  • Game 3 – 13% for a team up 2-0; 19% for a team tied 1-1; 12% for a team down 0-2.
  • Game 4 – 6% for a team up 3-0 or down 0-3; 19% for a team up 2-1 or down 1-2.
  • Game 5 – 12% for a team up 3-1; 25% for a team tied 2-2; 13% for a team down 1-3.
  • Game 6 – 25%

NOTE: The asymmetries in the list above are relics of the rounding of values in the table. In reality, the 12s and 13s in Games 3 and 5 are all equal, as are the 15 and 16 in Game 2.

Therefore, the highest impact wins in order are–

  • 50% – Game 7
  • 25% – Game 6; or Game 5 if tied 2-2
  • 19% – Game 4 in a 2-1 or 1-2 series; or Game 3 if tied 1-1

And naturally, the above scenarios also describe the highest impact losses.

The real world

It bears repeating that the above modeling is all based on two unrealistic assumptions–

  • The two teams are equally matched.
  • Home field/court/ice advantage does not exist.

 

Nonetheless, the approach taken here provides at least a baseline for understanding the probabilities associated with a seven-game series and provides a starting point for more advanced modeling.

 

 

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