Conditional probability and the length of a championship series in baseball,basketball,and hockey

1 Previous research

Only a few studies addressing the issue of the distribution of games have appeared in the scholarly literature. Mosteller (1952) examines the question of whether a seven-game World Series is sufficient for identifying the so-called “better” team. He derives a maximum likelihood estimates of a constant probability of one team winning each game of p ˆ = 0.65 , implying a corresponding probability of the other team winning of 1 - p ˆ = 0.35 . This result would seem to be inordinately high, implying that the average series is a mismatch. At first glance, his results might seem to be dominated by the period covered by his data of 1905–1951, in which perhaps a small number of powerful teams prevailed. As is well known, the New York Yankees have been a World Series powerhouse. In the 1905–1951 period covered in his study, the Yankees won about 38% of American League pennants. Nonetheless, in our sample, which covers 1923–2018, the Yankees have performed even better, winning 42.7% of American League pennants. But in later years, there are far more teams and, thus, there could be greater balance in the data. 1

Groeneveld and Meeden (1975) examine data from the World Series (1903–1973), NBA (1950–1973), and National Hockey League (1939–1967) with the objective of obtaining a single best probability estimate. They obtain maximum likelihood estimates of a constant probability of p ˆ = 0.58 for baseball, 0.654 for hockey, and 0.533 for basketball. While the result for baseball seems slightly one-sided, the finding for hockey suggests that on average, the Stanley Cup is an extraordinary mismatch.

Using World Series data from 1903–2005, Cassuto and Lowenthal (2007, 2011) examine the issue from an economic perspective. They argue that those with an economic interest in the event should be concerned about the likelihood of the event lasting four, five, six, or seven games. Their examination is couched in the following manner. Consider the multinomial X = (X₄,  X₅,  X₆,  X₇) where X _i is the number of games the series lasts, i = 4,  ...  7. Cassuto and Lowenthal use the chi-square test with three degrees of freedom to examine the null hypothesis that p ˆ = 0.50 maximizes the likelihood of the observed relative frequencies of X _i games. Using data from 1903–2003 (their 2007 study) and updating it with data from 1903–2005 (their 2011 study), they cannot reject the null hypothesis of p ˆ = 0.50 at the 0.05 level but are able to reject at the 0.10 level. They then find that the probability that minimizes the chi-square statistic is p ˆ = 0.569 . They acknowledge that relative strength and home field advantage could affect their conclusions, a point we take up later. Interestingly, they find that while there is a home field advantage in a given game, there is no home field advantage in the series as a whole. 2

The constant probability model has also been in used in other studies directed at related but different sports issues. For example, Ben-Naim et al. (2012) employ a constant probability model to examine the efficacy of single-elimination tournaments in contrast to each team playing each other team. The blogosphere has also contained discussion about the probabilities of the number of games. 3 Although several of these previous papers and discussions raise the issue that the probabilities may not be constant, all generally work with the independent constant probability assumption or leave the matter for others to resolve. In short, the research on this question has not proposed, what more examined, an alternative to the model of constant probability and independence.

In an excellent treatment of the mathematics of baseball, Ross (2007) shows how the ex ante distribution of games is determined, and he acknowledges that a probability of 0.5 does not fit the data very well. He states, however, that it is difficult to determine any other probability. He discusses the possibility of a conditional probability model but only in the context of a model in which one team is the favorite. For example, a favorite would be defined as a team having a probability of winning a game of more than 0.5. He then examines the formulas for the probability of a series lasting a certain number of games given that the favorite has already won or lost some games. Our work extends the idea of conditional probability by using historical data, but in our model any so-called favorite, could, for example, be the favorite after one or two games and the underdog later. We let the historical data tell us when these probabilities change.

In this study, we formalize the application of conditional probability to the model of seven-game championship series and thereby fill a much needed space in the evolution of formal models of these events. 4

2 Specification of the problem

If we begin with the assumption that an arbitrary team, say team A, has a probability p of winning each game, it is a simple matter to determine the probability that a series will last four, five, six, or seven games. 5 Denoting those probabilities as Pr(i), the probability that a series will last i games where i = 4, 5, 6, or 7, we have Pr ( 4 ) = p 4 + ( 1 - p ) 4 Pr ( 5 ) = 4 ( p 4 ( 1 - p ) + p ( 1 - p ) 4 ) Pr ( 6 ) = 10 ( p 4 ( 1 - p ) 2 + p 2 ( 1 - p ) 4 ) Pr ( 7 ) = 20 ( p 4 ( 1 - p ) 3 + p 3 ( 1 - p ) 4 ) .

The general formula is Pr ( n ) = ( n - 1 ) ! 3 ! ( n - 4 ) ! ( p 4 ( 1 - p ) n - 4 + p n - 4 ( 1 - p ) 4 ) .

Under the naïve assumption that p = 0.5, Table 1 shows these probabilities. They are, of course, well-known and documented in the literature. The expected number of games is as follows: E ( G ) = 4 ( 0.125 ) + 5 ( 0.25 ) + 6 ( 0.3125 ) + 7 ( 0.3125 ) = 5.81

As we will show, however, these probabilities and hence the constant probability assumption is a poor match to the observed relative frequencies, especially for certain sports.

We collect data for baseball’s World Series, basketball’s NBA Finals, and hockey’s Stanley Cup finals, the latter hereafter referred to simply as the “Stanley Cup.” The World Series began in 1903 but followed an inconsistent format until 1923. In some cases the Series was based on a best-of-seven structure, in others a best-of-nine structure, and in some years, tied games were permitted. Since 1923 however, the series has been played with a consistent best-of-seven format with tied games extending into extra innings until a winner is determined. Thus, we collect data starting in 1923 through 2018 from www.baseballreference.com. There are 551 World Series games played during that period.

The National Basketball Association was founded in 1949 as a successor to the Basketball Association of America. Its 1949–1950 season was played with 15 teams divided into three divisions. From the 1950–1951 season forward, it has been divided into Eastern and Western Divisions, which in some years are called “Conferences,” and in later years further broken into sub-divisions. We start with this second season of the NBA’s history, 1950–1951, and collect data through the 2017–2018 season from www.basketballreference.com. 6 There are 392 games played in the Championship finals during that period.

The National Hockey League was founded in 1917, and the league structure has changed many times. Not until 1968 did the league divide itself into eastern and western divisions. In 1975 it then realigned into the Clarence Campbell and Prince of Wales Conferences, each with three divisions. Nonetheless, these divisions did not compete with each other in the playoffs until 1982. In other words, until 1982, two teams from the same conference could end up competing in the finals for the Stanley Cup. In 1994 the structure was changed again, this time into eastern and western conferences, whereby the Stanley Cup finals would match the playoff champions of the eastern and western conference. For the hockey data, we start with 1939, because that was the first year the Stanley Cup followed a best-of-seven format that it has retained since that time. The data sources are www.hockey-reference.com and www.nhl.com and go through the 2017-2018 season. After accounting for the cancelation of the 2005 NHL season, there are 403 Stanley Cup finals games played during that period.

Table 2 provides descriptive statistics of the 95 World Series, 68 NBA Finals, and 79 Stanley Cup Finals in the data set. Although not shown in the table, a total of 36 teams have appeared in the World Series, 30 in the NBA Finals, and 28 in the Stanley Cup Finals. 7 Of course, the New York Yankees have been a dominant team in baseball, appearing in 38 series, winning 27. The second most successful team has been the St. Louis Cardinals, who have appeared in 19 series, winning 11. The Pittsburgh Pirates have won four of five series, for the best winning percentage with four or more series. The most unsuccessful team that has competed in at least one series would be the Brooklyn Dodgers who appeared in seven series, winning only one (1955).

The NBA has been dominated by the Boston Celtics and Los Angeles Lakers. The Celtics have appeared in 21 Finals, winning 17. The Los Angeles Lakers have appeared in 25 Finals, winning 11, and the Laker franchise also appeared in four other finals, winning three, as the Minneapolis Lakers. Two other very successful franchises are the Chicago Bulls, who have won all six of their appearances, and the San Antonio Spurs, who have won five of six. The worst record for a team that has appeared at least four times is the Cleveland Cavaliers who have won one of five series.

The most successful hockey teams have been the Montreal Canadians, Detroit Red Wings, Boston Bruins, and Toronto Maple Leafs. Montreal has won 20 of 27 Stanley Cup finals, while Detroit has won 9 and lost 12, and Boston has won five and lost 11. The Toronto Maple Leafs have appeared 14 times, winning 10, and the Chicago Black Hawks have won four and lost six. Of course, these are the oldest teams in the league, and thus, they have had more opportunities and less competition.

The table also shows some data on the home-away record. The home advantage can be interpreted several ways. One is the obvious notion of a home advantage for a given game. The team at home has won 56.20% of World Series games, 61.73% of NBA Finals games, and 60.56% of Stanley Cup Finals games. An alternative notion of home advantage is that the team that plays more home games in the series has an advantage. Under this definition of home advantage, the team with the edge has won 45.61% of World Series championships, 66.67% of NBA Championships, and 75.68% of Stanley Cups. 8

Of the two competing teams, the one with the better season record has won 50.57% of World Series, 72.88% of NBA Championships, and 77.46% of Stanley Cups. These results shed some light on whether many of these series have been mismatched, meaning that there is simply a stronger team. That could be the case in hockey as well as basketball, but it does not appear to be the case in baseball.

The last section of the table shows the various home-away formats used by the leagues. Baseball has followed the 2-3-2 format for 91 of 95 series, with two games in one city, the next three in the other city, and the final two in the first city. Basketball has used four different formats with 31 following the 2-3-2 format and 33 following the 2-2-1-1-1 format in which the last three games alternate cities. Hockey has used the 2-2-1-1-1 format for 74 of 79 series. 9

Table 3 shows the observed frequencies of four-, five-, six-, and seven-game series along with the expected frequencies under on the naïve assumption of a 0.5 probability for each game. As can be seen, there have been far more four- and seven-game World Series, and correspondingly fewer five- and six-game series that would be expected if in each game, each team had a 50% chance of winning. The average number of games, however, at 5.80 is almost precisely equal to the expected number of 5.81. This finding immediately makes us believe that 0.5 is the appropriate parameter. But, we have fit only the mean of the distribution. The assumption of a 0.5 probability is not very accurate, as indicated by the fact that the chi-square goodness of fit statistic, χ 2 = ∑ cells ( observed - expected ) 2 expected , rejects the constant probability null hypothesis with a p-value of less than 4%. This result is not surprising. Two distributions can certainly have the same or very close means and yet be quite different.

Interestingly, the NBA data come much closer to fitting the p = 0.5 model. The number of five-game series is precisely the theoretical value and the other observed and expected frequencies are somewhat close to their theoretical values, and the average number of games, at 5.76, is close to the expected number of 5.81. Nonetheless, the null is still rejected with a Type I error of only 6.2%. In addition, as we will show later, it is possible to obtain a better fit for NBA data and plenty of reasons to prefer one.

For the Stanley Cup, the differences between the observed and the expected frequencies are quite high. The percentage of four-game series is more than twice the expected number. This disproportionate result comes mostly at the expense of seven-game series, with about 20% of Stanley Cup finals going to seven games, while the expected percentage is slightly more than 31%. The average observed number of games of 5.47 is notably lower than the expected number under the 0.5 hypothesis, and the chi-square test rejects the null hypothesis at better than the 1% level. Thus, the evidence suggests that a model with a constant probability of 0.5 does not fit hockey or baseball at all and probably not even basketball.

We may be able to salvage the constant probability model if we can find a probability other than 0.5 that fits the data reasonably well. Consider first the World Series, in which we have too many seven-game series in relation to the number we would expect if p = 0.5. We first might wonder if we can alter the constant probability model to better fit the data, but we show in Appendix A that if a constant probability is assumed, the probability of a seven-game series is maximized at p = 0.5. Since the observed number of seven-game series is more than the expected number at p = 0.5, any other value of p will certainly take the expected number of seven-game series even further away from the observed number.

In spite of this impossibility, Mosteller (1952) suggests two ways of fitting a single probability estimate to the data. First, he suggests matching the expected value with the average number of games played. 10 We do this for the World Series and obtain p ˆ = 0.5333 . Unfortunately, this probability still results in rejection of the null hypothesis that the distributions are the same under the chi-square goodness of fit test, with a test statistic of 8.54 and a Type I error of 3.61%. There may be hope for the NBA Finals and the Stanley Cup, however, as the expected number of seven-game series is a higher than the observed frequencies. Perhaps we can arrive at a constant probability more than 0.5 that fits the data better.

For the NBA Finals, we obtain p ˆ = 0.5671 , which does not result in rejection of the null hypothesis using the chi-square test. Nonetheless, we find the disconcerting result that that the naïve assumption of p = 0.5 and the optimized assumption of p = 0.5671 produce fairly close chi-square statistics, 0.41 vs. 0.32. The reason for this result will become clear shortly. For the Stanley Cup, we obtain an estimate of p ˆ = 0.6760 , a result that suggests that the average Stanley Cup series is a mismatch.

Mosteller also suggests a maximum likelihood approach to identifying a single constant probability. In other words, given the observed data, which probability maximizes the likelihood that the sample would have occurred? Following Mosteller’s approach, let p _L  (j)  = the probability that the team that loses the series wins j games, where j = 0, 1, 2, or 3. The multinomial probability of drawing the sample is ( ( N ( 4 ) + N ( 5 ) + N ( 6 ) + N ( 7 ) ) ! N ( 4 ) ! N ( 5 ) ! N ( 6 ) ! N ( 7 ) ! ) p L ( 0 ) N ( 4 ) p L ( 1 ) N ( 5 ) p L ( 2 ) N ( 6 ) p L ( 3 ) N ( 7 ) where N (i)  = the number of four-, five-, six-, and seven-game series that have occurred. Because the term in the large parentheses is fixed for a given sport, one can focus on the remaining term, p _L  (0) ^N(4)p _L  (1) ^N(5)p _L  (2) ^N(6)p _L  (3) ^N(7), which is the kernel of the multinomial probability. Maximization of this value results in maximization of the likelihood that the sample would have been drawn from a population in which the probability of a team winning a game is constant across the series and, thus, provides a maximum likelihood estimate. Figure 1 illustrates the graph of this multinomial kernel for the three sports.

OPEN IN VIEWER

Fig. 1

Maximum Likelihood Estimate of Constant Probability of One Team Winning the World Series, NBA Finals, and Stanley Cup. A. World Series. B. NBA Finals. C. Stanley Cup. The line is the value p _L  (0) ^N(4)p _L  (1) ^N(5)p _L  (2) ^N(6)p _L  (3) ^N(7), _,which is the kernel of the multinomial likelihood function where p _L  (j) is the probability that the team that loses the series wins j games (j = 0, 1, 2, or 3) and N(i) is the number of four-, five-, six-, and seven-game series that have occurred. The observed maxima identify the maximum likelihood estimates of a constant probability of one team winning.

For baseball, Fig. 1A, we see that the probability of observing 18 four-game series, 20 five-game series, 20 six-game series, and 37 seven-game series is maximized at roughly p ˆ = 0.43 and the complement, p ˆ = 0.57 . 11 Nonetheless, the variation over a wide range is relatively small. Moreover, a constant probability of 0.43 (0.57) also suggests that the average series is somewhat mismatched. In addition, to the maximum likelihood estimator, we examine the chi-square statistic over this range of probabilities. 12 It is significant at a p-value of 4% or better for all probabilities between roughly p ˆ = 0.4 and p ˆ = 0.6 . Thus, we know that we still do not have a good fit.

In Fig. 1B, for the NBA, we find yet another disconcerting result. While the maximum occurs at p ˆ = 0.44 with the complement of 0.56, all probabilities in the range of 0.43 to 0.57 have virtually the same likelihood. In addition, the chi-square is not significant over this entire range. Thus, we are led to the rather disconcerting conclusion that almost any probability over this range works about as well as any other.

For the Stanley Cup, Fig. 1C, we have maxima at around p ˆ = 0.32 and p ˆ = 0.68 , the same result previously found. Again, such lop-sided odds seem unrealistic.

Hence, for baseball, no single value of p seems appropriate. For basketball multiple values of p seem to fit the data. For hockey, extreme values of p best fit the data.

In this section we attempt to determine if we can improve the model of the number of games in best-of-seven championship series by incorporating conditional probability. Historical data can be tabulated based on conditions that exist to that point to provide estimates of conditional probabilities. Naturally we can always fit a model to historical conditional frequencies. The insight of this analysis is to reveal the extent to which the conditional probabilities vary across different conditions. If they do not vary much, then a conditional probability model will not be an improvement over a constant probability model.

Table 4 takes this approach by showing the relative frequencies with which teams win, given their previous wins in the series. The corresponding numbers in parentheses are the numbers of occurrences of the condition. Thus, for example, in the second line for the World Series one team has been ahead two games-to-none 46 times, and it has won the third game 43.48% of the time.

In baseball, we see that the team winning the first game has won the second game a little less than 49% of the time, and when it wins the second game, it wins the third game about 47% of the time. Yet if it wins the third game, it wins the fourth game and sweeps the series 90% of the time. A team ahead two- games-to-one wins the fourth game about 47% of the time, while a team ahead three-games-to-one wins the fifth game about 54% of the time. Perhaps most strikingly, a team ahead three-games-to-two wins the sixth game only about 35% of the time. In other words, the team with its back to the wall, down three games-to-two, wins almost two-thirds of the time to send the series into the seventh game. This phenomenon explains why the number of seven-game series is so large in relation to what is implied by the p = 0.5 model. We will explore this result in more detail later.

Turning to basketball, a team up one game-to-none wins the second game about 47% of the time. Being up two games-to-none, however, has been a much more difficult situation, with the team ahead winning only about 38% of the time. If the team is ahead two games-to-one, it wins game four about 48% of the time. As with baseball, a team ahead three games-to-none has a very high chance (75%) of a four-game sweep. Teams ahead three games-to-one and three games-to-two have won the next game about 57% and 55% of the time, respectively. In the latter case, with a team behind three games-to-two and winning the sixth game only about 45% of the time, we would expect a somewhat smaller number of seven-game series than would occur with the constant probability and independence model. And indeed, that is the case, as was shown in Table 3.

For the Stanley Cup, teams that win the first game go on to win the second game about 65% of the time. If a team is up two games-to-none, it wins the third game about 52% of the time. If it is up three-games-to-none it wins the fourth about 74% of the time. 13

To summarize, if the constant probability and independence model were correct, these conditional frequencies would be approximately the same. Some modest variation might be expected and indeed a number of these frequencies are close to 50%. But too many are far away from 50%. From this data, we can easily see that baseball has too many seven-game series and too few six-game series because teams behind three-games-to-two tend to win at far better than a 50% rate. It also has too many four-game series because teams ahead three-games-to-none sweep the series 90% of the time. Basketball fits the constant probability rule more closely than the other two sports, but it has a bit too few seven-game series and slightly more six-game series because teams up three-games-to-two win the sixth game at well more than a 50% rate. Hockey has far too many four-game series because teams up three-games-to-none win and sweep 75% of the time. It has too few seven-game series, because teams ahead three-games-to-two win the sixth game 60% of the time. In addition, the excessive number of four-game sweeps also reduces the number of seven-game series. These results are completely inconsistent with the constant probability model.

In addition, by using historical data we can fit the distribution exactly, without having to know the odds of one team winning when the series is tied. In fact, the probabilities when the series is tied are irrelevant as there is no pre-condition that favors one team. Hence, we can assume p_0,0 = p_1,1 = p_2,2 = p_3,3 = 0.5 Letp _k , m be the probability that a team will win the (k + m+1) ^th game, having previously won k games and lost m games. We first determine the probability that the series will last four games. The probability of an arbitrary team A winning the series in four games is p 0 , 0 p 1 , 0 p 2 , 0 p 3 , 0 = 0.5 p 1 , 0 p 2 , 0 p 3 , 0

The series could also end in four games if team A loses the first four games. If the conditional probabilities apply equally to each team, then the probability of team A losing all four games is 14 ( 1 - p 0 , 0 ) p 1 , 0 p 2 , 0 p 3 , 0 = 0.5 p 1 , 0 p 2 , 0 p 3 , 0

Thus, the probability that the series ends in four games is the sum of these probabilities, Pr ( 4 ) = p 1 , 0 p 2 , 0 p 3 , 0 .

Now consider the probability that the series will end in five games, Pr(5). By definition, Pr(5) is the probability that after four games, one team is up three-games-to-one and that team wins the fifth game. 15 There are eight possible ways that teams A and B can play four games and have one team win three and the other win one. Denoting the series-winning team as either A or B, the eight possible sequences of winners are AAAB, AABA, BAAA, ABAA, BBBA, BBAB, ABBB, and BABB. For the series to end on the fifth game, we must append to these eight sequences either A or B such that A or B has four wins. For team A to win, the outcomes would have to be AAABA, AABAA, BAAAA, and ABAAA. For team B to win, the outcomes would be BBBAB, BBABB, ABBBB, and BABBB. Appendix B shows that the probabilities of these events combine to

Pr ( 5 ) = p 3 , 1 ( p 1 , 0 p 2 , 0 ( 1 - p 3 , 0 ) + p 1 , 0 ( 1 - p 2 , 0 ) p 2 , 1 + ( 1 - p 1 , 0 ) p 2 , 1 ) .(1)

Now consider the probability that the series ends in six games. Given what we already know, this result is easy to obtain. First consider that the probability that the series lasts at least six games is 1 –Pr(4) –Pr(5). The probability that the series lasts exactly six games is the probability that it lasts at least six games times the probability that the team ahead three-games-to-two wins the sixth game. 16 Thus, Pr ( 6 ) = ( 1 - Pr ( 4 ) - Pr ( 5 ) ) p 3 , 2 .

And finally, the probability that the series lasts seven games is simply the complement of the probabilities we have obtained to this point, Pr ( 7 ) = 1 - Pr ( 4 ) - Pr ( 5 ) - Pr ( 6 ) .

Using this information and estimates of the conditional probabilities from the historical frequencies, we can match the distribution of games precisely. The results are obtained by building a binomial tree for the 70 sequences, inserting the probabilities of each path, and doing the appropriate multiplications. Results are verified using the equations above for Pr(4), Pr(5), Pr(6), and Pr(7). Table 5 presents these results. 17

The NBA results are particularly interesting in that an assumption of p = 0.5 throughout the series gave a reasonably good fit. But to make such an assumption is to believe that a team has a 50–50 chance of winning even when it is down three games-to-none, when the actual frequency of winning is only 25%, and two-games-to-none, when the actual frequency of winning is, 62.5%). While the remaining cases do not deviate notably from 50%, we see here that even incorporating conditional probabilities that are far away from 0.5 produces not only a good fit but a better fit. And in baseball, advocates for use of a probability slightly different from 0.5 fail to recognize that the frequency of winning when down three games-to-none is 10%, and when down three-games-to-two, the frequency of winning is 64.9%. In hockey, the conditions of being up one-game-to-none and three-games-to-none require probabilities far above 50%, specifically, 64.6% and 74.1%, respectively.

It is important to note that the relative frequencies do not do not constitute the only set of estimates of probabilities that will fit the distribution. We are fitting a multinomial X = (X₄,  X₅,  X₆,  X₇) where X _i the number of series ending in i games, but there are six parameters, p_1,0, p_2,0, p_3,0, p_3,1, p_3,2, and p₂, 1. With more parameters than constraints, there are clearly multiple combinations of the conditional probabilities that fit the data. Based on the fact that we are interested in explaining the empirical history of these three sports, we are fitting a distribution based on the ex post frequency of series ending in i games, and we have no further information on the ex ante probabilities, our statistical estimates are based on p ˆ m , k = numer of wins for team ahead m - to - k number of m - to - k situations .

It has been said that probability theory and the World Series do not agree, but such a statement is based on attempting to force the constant probability and independence model to fit the data. 18 As we show here, conditional probability provides a richer framework that better fits the data. With the large number of parameters in the model, agreement is guaranteed.

We have shown that we can fit the entire tree to the conditional probability model. But there remains the question of whether we can make the constant probability model work by evolving through time and resolving uncertainty as we get closer to the end of the series. To answer this question, we identify that there are 47 nodes in a best-of-seven series at which there is conditioning information, meaning the number of games won by each team to that point. We also require that at that point the series will continue at least one game. These requirements eliminate the time 0 node, as no previous games have been played. They also eliminate all nodes at which the series ends and the subsequent nodes that would otherwise evolve. And, they also eliminate game 6 nodes, because if we are at game 6 and the series is not over, there will definitely be a game 7 so there is no uncertainty over how many games the series will require..

At each of these conditioning nodes, we can compute the expected distribution of the series length. Because we use conditional frequencies as proxies for the conditional probabilities, the probabilities of four-, five-, six-, and seven-game series viewed from any node will equal the true frequencies from that point in time. Alternatively, we can also apply the constant probability model and determine the expected number of games at each node. As we evolve through the tree, picking up information that resolves uncertainty, we could find that the constant probability model begins to work better. In this manner, we are updating the constant probability model for the conditions that occurred throughout the tree. As noted, we can do this at each node, thereby obtaining updated distributions of the length of the series, which we can compare with the distributions based on the conditional probabilities, which fit the data perfectly. In this manner, we could find that the constant probability model improves as we roll through the tree. The results of this test are in Table 6, which summarizes the prob values of the chi-square test of the null hypothesis that the updated distributions from the constant probability model fit the data as of each time point from after game 1 to after game 5. If the constant probability model fits the data we should see few rejections and the propensity to reject decreasing as we move forward in time.

For baseball (Panel A), we do see some improvement in that after the fourth game, half of the nodes fail to reject but this trend does not carry over to the fifth game. With basketball (Panel B), we see a definite propensity to not reject as we evolve. Recall from Table 3, that the constant probability model did provide a somewhat reasonable fit to the data, but the problem with the NBA data was that a wide range of probabilities seemed to fit. In the Stanley Cup results (Panel C), the updating process also shows a tendency to improve the constant probability mod as we tend to not reject as we evolve.

These results are not surprising. They show that as uncertainty is resolved, simpler models often do improve. But if one uses the simpler model, little would be gained and considerable information would be ignored. And obviously, the model does not help us much at all before the first game.

In this section we address the issues of momentum and home advantage as well as the interpretation of the concept of being evenly matched. We attempt to determine if these factors can explain the variation in the conditional probabilities.

8 Conclusions

Attempts to assign a single probability of victory to a best-of-seven series suffer from the problem that no particular probability is satisfactory in explaining the World Series, a wide range of probabilities seem to provide similar results for basketball, and a relatively lop-sided probability works best for hockey. The games in these championships series are simply not independent experiments. Yet, the research on this subject has never formalized the next step of showing how we can build a better model. In this paper, we propose that the use of conditional probabilities provides a better fit in explaining the distribution of games. We show that using the probabilities of victory conditioned on the number of games won to that point enables us to fit the distribution perfectly. Of course, it is tautological that we can fit empirical frequencies and fully explain outcomes, but the insight of these results is in the fact that these conditional probabilities vary widely within a sport, across sports, and deviate substantially from the constant probability model.

There are multiple solutions for the estimates of the conditional probabilities, including ex ante estimates. As with any probability distribution, one can estimate based on ex ante values, use historical estimates, or employ a combination of the two. The interest in sports statistics is largely focused on explaining the past. Hence, the use of historical relative frequencies is an appropriate way to estimate the conditional probabilities. In doing so, we are able to examine the effects of momentum and home advantage, neither of which is shown to have much of an impact on these conditional probabilities. Certainly there is a home advantage in all sports, but the unusually high winning percentages in certain games in championship series are not explained by a normal home field advantage. Moreover, momentum, while possibly existing in hockey, is still a questionable notion in an event that amounts to but a short series. And based on the research of others, the whole notion of momentum and streaks in sports has been highly discredited in a number of studies. In all likelihood, any resemblance between consistently strong performance and momentum is likely just a mismatch.

Left without momentum or home advantage, the variation in conditional probabilities is probably best explained by the mere fact that the odds tilt toward one team in certain situations. One in particular is worth repeating. Baseball teams behind three-games-to-two rally to win about 65% of the time in game six. Behind three-games-to-none, however, they win only 10% of the time. These statistics vary widely across the three sports. Further research may shed some light on why such results are observed.