### How likely is it that a series will go to seven games if the teams are evenly matched?

#### How likely is it that a series will go to seven games if the teams are evenly matched?

##### Jean-Patrice Martel
###### Posted June 22, 2014

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You often hear the prediction: "the teams are of almost equal strength, so this series will probably go to the limit". Certainly, judging from this year's (2014) playoffs, where nearly half the series went to the limit, and where everyone says there is more parity than ever, it is certainly tempting to conclude that the saying is right. But is there any scientific basis to predicting that a best-of-seven series opposing two well-matched teams is more likely to go to the limit of seven games?

Let us assume a situation where the two teams are exactly equal in strength. Say that two friends agree to a best-of-seven series of coin tosses. Assuming that their coin is perfectly balanced and the "flipper" does not cheat, we can probably safely say that both teams are perfectly even in strength.

In such a case, the probability that the best-of-seven series reaches the limit of seven games is 31.25 %, exactly the same as the probability that the series will end in six games. It is also possible that, despite the perfect equality of forces, the series will end after four consecutive wins of one team or the other (probability of 12.5 %) or after five games (25 % probability). So even if the teams ("heads" and "tails") are perfectly equal in strength, the probability that the series will end in less than 7 games is 68.75 %, or more than two thirds. And the probability that it will end in four or five games is higher (37.5 %) than the probability that it will end in seven (31.25 % as mentioned).

Of course, one could argue that even though the teams involved are of equal strength, there is still the home ice advantage. It is generally agreed that the team that plays at home has an advantage over the visiting team, and that could be part of the reason why many believe that the series is more likely to go to the limit of seven games than to end in six or fewer.

That is indeed a good point, which can be demonstrated easily by taking an extreme example: suppose that while the teams are theoretically of equal strength, the home-ice advantage is so important that the team playing at home has a 100 % chance of winning. In this case, the team with the home-ice advantage in a best-of-seven NHL series will always win the series, with wins in games 1, 2, 5 and 7 (and the NHL will have much bigger worries than the fact that there are 16 teams in the Eastern Conference but only 14 in the Western Conference). That said, since the 2004/05 NHL lockout, there has only been one series out of 135 in which the home team won all games: the 2013 Conference semi-final series between San Jose and Los Angeles.

Of course, in reality, the probability that the team that plays home will win is not 100 %, far from it. In the playoffs since the lockout of 2004/05, up to and including the 2014 playoffs, the team playing at home has won 56.28 % of its games (to be clear, we are talking about games, not series), with a "peak" of 68.60 % in 2013 and a low just the previous year, in 2012, of 45.35 %. Yes, in the playoffs of 2012, the home team won fewer games overall than the visiting team. Any other year since the lockout had an at home success rate higher than 50 %, but we note that in 2010 it was only 51.69 %.

So what is the probability that a series ends in seven games if the home team has a 56.28 % chance of winning each game, rather than the 50 % used thus far? Rest assured: the probability that the series will end in seven games is indeed higher than the probability that it will end in six games. Here are the odds:

Seven games: 31.55 % (and in that case, the team starting the series at home, such as the 2014 Kings, has a 17.76 % probability of winning the series, while the team that started the series on the road, e.g. the 2014 Rangers, has a 13.79 % probability of winning the series).

Six games: 31.35 % (17.24 % for the 2014 Rangers against 14.10 % for the 2014 Kings – so in fact, if the series ends in 6 games, it is more likely that the winner will be the team that started the series on the road).

Five games: 24.99 % (14.07 % for the Kings against 10.93 % for the Rangers).    Four games: 12.11 % (6.05 % for each team).

The probability that the series ends in seven games is therefore only very slightly higher than the probability that it ends in six: 31.55 % against 31.35 %. If someone favours the team that has home ice advantage for the series, then the probability that it will win is as follows:

• In seven games: 17.76 %
• In six games: 14.10 %
• In five games: 14.07 %
• In four games: 6.05 %

For someone favouring the team that does not have home ice advantage for the series, the probability that it will win is as follows:

• In seven games: 13.79 %
• In six games: 17.24 %
• In five games: 10.93 %
• In four games: 6.05 %

At this point, the reader might say: "This is all very nice, but mathematical probabilities and real life are two different things." However, it looks like real life is siding with mathematical probabilities. Below, the second column shows the probability of winning the series based on a 50-50 chance of winning any given game. The third column shows the probability of winning the series based on a 56.28% chance of the home team winning any game. And the fourth column shows the actual percentage of series won in the given number games since the 2004/05 lockout.

# gamesPred. 50%Pred 56.28%Actual
4 games 12.50% 12.11% 12.59%
5 games 25.00% 24.99% 27.41%
6 games 31.25% 31.35% 30.37%
7 games 31.25% 31.55% 29.63%

Not only are the actual figures extremely close to the ones predicted by mathematical probabilities, but the number of 7-game series, despite having had so many in 2014, is still actually slightly lower than the predicted one, while the number of 4-game series is (also very slightly) higher than the predicted one.

We also note with interest that even though the team playing at home has a probability of 56.28 % of winning any given game, this only translates into a 51.98 % probability for the team having the home ice advantage for the entire series to win the series. Pushing the parenthesis even further, if the probability of the home team winning any given game was 75%, this would still only translate into a 59.40% probability of winning the series for the team having home-ice advantage. That said, in this case the reality shows that the team with "overall" home ice advantage fared a little better than its predicted probabilities: 55.56% of all series (against the predicted 51.98%) were won by the team with "overall" home ice advantage.

But what about those cherished game sevens? One might point out the difference between general probabilities and specific probabilities: the "general" likelihood that the home team wins a given game cannot be compared to the "specific" probability that the home team will win game seven of a series.

Indeed.

What were the results since the 2004/05 lockout of playoff game sevens in the NHL?

Home team: 18 wins

Visiting team: 22 wins

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