How Many Games Does it Take to Win the World Series of Major League Baseball?

Taishi Kim
Maki Takeda
Takashi Ikeda
Tomoaki Shintani
Ryohei Miyadera

Abstract.

The World Series of MLB consists of seven games. If a team wins four out of the seven games, the team becomes the World Champion. If a team has already won four, the rest of the game will be cancelled. Some people believe that the World Series of MLB has been fought between almost evenly matched teams. By comparing the statistics of World Series of MLB and mathematica simulation we can prove that this belief is false. The Japan Series has the same kind of system to choose the champion, and the statistics of it shows that it has been fought between almost evenly matched teams. The contrast between the statistics of the World Series and the Japan Series is very interesting.

We are the first people who found this fact. We presented our idea at the International Mathematica Symposium on July 8 in 2003.

In this article we write Mathematica code in blue. You can read this article without the knowledge of Mathematica, but Mathematica has been a very useful tool for us to study this theme.

If you are going to do the calculation in this article, please load the following package. You will get an error if you do some calculations in this article without loading the package. If you get the error, it is better to quit Mathematica and start it again.

<< Graphics`MultipleListPlot`

Introduction.

The World Series consists of seven games. If a team wins four out of the seven games, the team becomes the World Champion. If a team has already won four, the rest of the game will be cancelled. The expected number of games is 5.9 in the history of the World Series compared to 5.8125 of the expected value of mathematically calculated probability for evenly matched teams. These are well known and correct facts.

From this data many people reached the conclusion that the World Series has been fought between almost evenly matched teams. For example Prof.Paul J.Nahin [1] expressed almost the same opinion in his famous book published in 2000.

We could prove that this conclusion was very far from the truth by using computer simulations.

We were the first people who studied this fact about the World Series, and presented our idea on July 8, at IMS2003. Please see Miyadera [2]. Shortly after that some people began to be interested in the same problem. For example professors B. Stein and M. Heil, [4]of American Institute of Physics wrote about it.

Stein and Heil wrote that the frequency of 7 game World Series in the history is a lot more than expected by probability, and the probability of the occurrence of that kind of thing is very small statistically.

What B. Stein and M. Heil wrote was almost the same as we presented at IMS2003, but we also studied 4 game World Series, too. There are too many 4 game World Series in its history.

We are also going to talk about the Japan Series which is the championship in Japan. As to the Japan Series, the distribution of probabilities of 4, 5, 6 and 7 games is almost the same as that of the Series  between almost evenly matched teams. This fact is well known in Japan, but the difference between the statistics of the World Series and the Japan Series is very interesting.

In this article we used simple computer simulations instead of the theory of statistics. This is a very good way, because most high school students can understand the argument.

These facts were discovered by high school students, and their teacher (Miyadera) helped them to do computer simulations.

1. Probability theory of the world series.

We are going to study the basic theory of probabilities of the World Series. These are well known facts, but they are very important.

Theorem 1. Two teams A.B are going to fight for the championship of the World Series. We suppose that the team A wins with the probability of x every time it play against the team B.

(1) The probability that the World Series will require 4,5,6 or 7 is (x^4)+(((1 - x))^4), 4(x^4)(1-x) + 4(((1 - x))^4)x, 10 (x^4)(((1 - x))^2)+ 10 (x^2)(((1 - x))^4), 20 (x^4)(((1 - x))^3)+ 20 (x^3)(((1 - x))^4).

(2) The average length of the games is 4 + 4x+ 4(x^2)+ 4(x^3)–52(x^4)+ 60(x^5)– 20(x^6).

Proof. First we are going to calculate the probability that the World Series require 4 games. We represent the situation of the team A's winning four games in straight by AAAA, and its probability is (x^4). Similarly the probability for BBBB is (((1 - x))^4). Therefore the probability that the World Series requires 4 games is (x^4)+(((1 - x))^4).

The next is the probability of 5 games. The possible outcomes of the game are AAABA,AABAA,ABAAA,BAAAA and ABBBB,BABBB,BBABB,BBBAB. Therefore the probability of 5 games is  4(x^4)(1-x) + 4(((1 - x))^4)x.

The next is the probability of 6 games. The possible outcomes of the game are AAABBA, AABABA,  AABBAA, ABAABA, ABABAA, ABBAAA, BAAABA, BAABAA, BABAAA, BBAAAA and AABBBB, ABABBB, ABBABB, ABBBAB, BAABBB, BABABB, BABBAB, BBAABB, BBABAB, BBBAAB. Therefore the probability of 6 games is 10 (x^4)(((1 - x))^2)+ 10 (x^2)(((1 - x))^4).

Similarly we can calculate the probability of 7 games. It is 20 (x^4)(((1 - x))^3)+ 20 (x^3)(((1 - x))^4).

The average of the number of the games required is  4×((x^4)+(((1 - x))^4)) + 5×(4(x^4)(1-x) + 4(((1 - x))^4)x) + 6×(10 (x^4)(((1 - x))^2)+ 10 (x^2)(((1 - x))^4)) + 7×(20 (x^4)(((1 - x))^3)+ 20 (x^3)(((1 - x))^4)) = 4 + 4x+ 4(x^2)+ 4(x^3)–52(x^4)+ 60(x^5)– 20(x^6).

We can calculate these probabilities and average by Mathematica.

Although it is not difficult to get these probabilities by hands, this kind of calculation shows the power of Mathematica. Later in this article we are going to use the power of Mathematica to calculate more complicated probabilities in Theorem 2.

Remark. Here we use the function st[ ]. This function is the same as the function Strings[ ] defined in Pemmaraju and S. Skiena [3]. We did not use the name "Strings", because Mathematica has a built-in function Strings in the Package "DiscreteMath`Combinatorica". You can use this built-in function, but for some unknown reason we got an error message if we try to use the package "DiscreteMath`Combinatorica" and "Graphics`MultipleListPlot`" at the same time. In fact you can use both if you neglect the error message, but we are afraid that the message may confuse many people, so we decided not to use the package "DiscreteMath`Combinatorica".

Clear[p, q, x];
st[l_List, 0]  :=  { {} }
st[l_List,  k_Integer]  :=  
      st[Union[l],  
          k]  /;  (Length[l]  =!=  Length[Union[l]])
st[l_List, k_Integer]  :=  Distribute[Table[l,  {k}],  List]; 
    probability[n_] := 
      Block[{a1, b, c, f}, 
        a = Select[st[{p, q}, n], 
            Count[#, p] == 4 && Part[#, n] == p || 
                Count[#, q] == 4 && Part[#, n] == q &]; 
        f[x_List] := Apply[Times, x]; b = Map[f, a]; 
        c = Apply[Plus, b]];
t = Table[probability[n], {n, 4, 7}];
p = x; q = 1 - x; 
    average = Simplify[t . {4, 5, 6, 7}];
Print[t]; Print[average];

Example 1.  Here are graphs of the probabilities.Green for 4 games,red for 5,blue for 6 and yellow for 7.

\(It\ is\ natural\ to\ assume\ that\ the\ probability\ x\ is\ near\ \ 0.5 . \ Under\ this\ assumption\ we\ are\ going\ to\ study\ the\ following\ Figure\ 1. \)

\(If\ x\  = \ 0.5, \ 
    then\ by\ Figure\ 1\ we\ know\ that\ the\ probability\ of\ 6\ games\ and\ 7\ games\ are\ the\ same . \ If\ not, \ 
    the\ probability\ of\ 6\ games\ is\ \(\(bigger\)\(.\)\)\)

Fig. 1  The probabilities of 4, 5, 6 and 7 games.

Mathematica calculation to make Fig 1.

I used two steps to create the above graphics. I do not know a better method to create it.

ff = 
        Plot[{4 ((1 - x) (x^4) + (((1 - x))^4) x), 
            (x^4) + (((1 - x))^4), 
            10 ((((1 - x))^2) (x^4) + (((1 - x))^4) (x^2)), 
            20 ((((1 - x))^3) (x^4) + (((1 - x))^4) (x^3))}, {x, 0, 
            1}, PlotStyle -> {{RGBColor[1, 0, 0]}, {RGBColor[0, 1, 
                  0]}, {RGBColor[0, 0, 1]}, {RGBColor[1, 1, 
                  0]}}];
Show[{ff, {Graphics[{{RGBColor[0, 1, 0], 
                Text[4  games, {0.97, 0.7}]}, {RGBColor[1, 0, 0], 
                Text[5  games, {0.95, 0.28}]}, {RGBColor[0, 0, 1], 
                Text[6  games, {0.85, 0.18}]}, {RGBColor[1, 1, 0], 
                Text[7  games, {0.74, 0.08}]}}]}}, 
      Background -> GrayLevel[0.7]]

Remark. Although it is easy to make this graph with software like Mathematica, we have never seen this kind of graph in any articles and books on the World Series Problem. I learned this graph from Mr. Chiboshi who was a math teacher at Kwansei Gakuin high school 15 years ago.

2. The real data from the statistics of the Series.

2.1 Real data from the Japan Series.

In Japan two teams from two different league meet at the Japan Series for the championship, and the rule to choose the champion is the same as that of the World Series.

The following is the real data from the 56 year's history of the Japan Series. The number of games for each years is presented as a list below.

japanseries={6,5,6,6,7,7,6,4,7,4,4,6,6,7,7,5,6,6,6,6,5,5,5,5,6,4,7,5,7,7,7,6,6,7,7,6,7,6,5,7,4,7,7,7,6,5,5,5,6,5,6,5,4,7,7,4}

From the data we made the following table.

({{number of games, 4, 5, 6, 7, average}, {percentage , (7/56) = 0.125, (13/56) = 
            0.232, (18/56) = 0.321, (18/56) = 0.321, 5.839}})

Table 1  Statistics of the Japan Series.

2.2 Real data from the World Series.

TheWorld Series started in 1903 and we have the data up to 2005, but they played under a different rule in 1903, 1919, 1920 and 1921. They did not have one in 1904 and 1994.

The following is the data from 1905 to 2005.

worldseries97={5,6,4,5,7,5,6,7,5,4,5,5,6,6,4,6,7,7,7,4,4,5,6,7,4,5,7,6,6,5,4,4,7,5,5,5,6,7,7,7,6,5,4,6,7,6,4,7,7,7,7,6,7,5,7,4,7,7,4,7,7,5,5,7,7,7,5,7,4,6,6,7,6,6,7,5,5,7,7,7,5,4,4,7,6,6,6,6,7,4,4,5,7,7,6,4,4}

({{number of games, 4, 5, 6, 7, average}, {percentage , (19/97) = 0.196, (21/97) = 
            0.216, (22/97) = 0.227, (35/97) = 0.360, 5.753}})

Table 2  Statistics of the World Series from 1905 to 2005.

We also made the data of the last 50 years. They did not have the Series in 1994, so we have a list of 49 years.

worldseries49={7,7,7,7,6,7,5,7,4,7,7,4,7,7,5,5,7,7,7,5,7,4,6,6,7,6,6,7,5,5,7,7,7,5,4,4,7,6,6,6,6,7,4,4,5,7,7,6,4,4}

({{number of games, 4, 5, 6, 7, average}, {percentage , (9/49) = 0.184, (8/49) = 
            0.163, (10/49) = 0.204, (23/49) = 0.469, 5.94}})

Table 3  Statistics of the World Series from 1956 to 2005.

We are going to refute the conclusion presented by Prof.Paul J.Nahin, so we have to use the same data that he used in his book Nahin [1]. This is the list of 91 years, since he had the data up to 1999.

worldseries91={5,6,4,5,7,5,6,7,5,4,5,5,6,6,4,6,7,7,7,4,4,5,6,7,4,5,7,6,6,5,4,4,7,5,5,5,6,7,7,7,6,5,4,6,7,6,4,7,7,7,7,6,7,5,7,4,7,7,4,7,7,5,5,7,7,7,5,7,4,6,6,7,6,6,7,5,5,7,7,7,5,4,4,7,6,6,6,6,7,4,4}

({{number of games, 4, 5, 6, 7, average}, {percentage , (17/91) = 0.187, (20/91) = 
            0.220, (21/91) = 0.231, (33/91) = 0.363, 5.769}})

Table 4  Statistics of the World Series in Paul J.Nahin [1].

Remark. Please note the fact that the averages are almost the same in Tables 1, 2, 3, 4 and 5. Perhaps many people thought that this fact alone would give them enough reason to assume that probabilities are almost the same.

3. Comparison between the real data and the mathematical theory .

If we put x = 0.5 in Theorem 1, then we get the following table. This is the mathematical probabilities for evenly matched teams.

({{number of games, 4, 5, 6, 7, average}, {probability, 0.125, 0.25, 0.3125, 0.3125, 5.813}})

Table 5  Mathematical probability of the Series for evenly matched teams.

3.1 Comparison between the statistics of the Japan Series and the mathematical theory.

Example 2. We are going to compare Table 1 and Table 5.

Here are graphs of the probabilities for the number of the games.  Green is the probabilities based on the mathematical theory, and  red  is the probabilities based on the history of the Japan series. The similarity is obvious.

From this data it is very reasonable to reach the conclusion that the Japan Series has been fought between almost evenly matched teams.

Fig. 2  The Japan Series and the mathematics theory.

Mathematica calculation to make Figure 2.

If you get some kind of error message here, please load the package again.

I used two steps to create the above graphics. I do not know a better method to create it. After I made the graphics, we retouched it with graphic software.

japanseries = {6, 5, 6, 6, 7, 7, 6, 4, 7, 4, 4, 6, 6, 7, 7, 5, 6, 6, 
          6, 6, 5, 5, 5, 5, 6, 4, 7, 5, 7, 7, 7, 6, 6, 7, 7, 6, 7, 6, 5, 7, 
          4, 7, 7, 7, 6, 5, 5, 5, 6, 5, 6, 5, 4, 7, 7, 
          4};
data1 = 
        Table[{n, Count[japanseries, n]/Length[japanseries]}, {n, 4, 
            7}];
x = 1/2; 
    data2 = {{4, (((1 - x))^4) + (x^4)}, {5, 
          4 (((1 - x))^4) x + 4 (1 - x) (x^4)}, {6, 
          10 (((1 - x))^4) (x^2) + 10 (((1 - x))^2) (x^4)}, {7, 
          20 (((1 - x))^4) (x^3) + 20 (((1 - x))^3) (x^4)}}; 
    gg = MultipleListPlot[data1, data2, 
        PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, 
        PlotJoined -> True, Frame -> False, 
        PlotRange -> {0, 0.5}, 
        Background -> GrayLevel[0.7]];
gg2 = 
      Show[{gg, {Graphics[{{RGBColor[0, 1, 0], 
                  Text["mathematics theory", {4.8, 0.3}]}, {RGBColor[1, 
                    0, 0], Text["Japan series", {5.5, 0.2}]}}]}}, 
        Background -> GrayLevel[0.7]]

3.2 Comparison between the statistics of the World Series and the mathematical theory.

Example 3. We are going to compare Table 2 and Table 5.

Here are graphs of the probabilities for the number of the games.  Green is the probabilities based on the mathematical theory, and  red  is the probabilities based on the history of the world series in Table 2.

The difference is obvious. From this data it is very difficult to reach the conclusion that the World Series has been fought between almost evenly matched teams.

Mathematica calculation to make the above figure.

I used two steps to create the above graphics. I do not know a better method to create it. After I made the graphics, we retouched it with graphic software.

worldseries97 = {5, 6, 4, 5, 7, 5, 6, 7, 5, 4, 5, 5, 6, 6, 4, 6, 7, 7, 
        7, 4, 4, 5, 6, 7, 4, 5, 7, 6, 6, 5, 4, 4, 7, 5, 5, 5, 6, 7, 7, 7, 6, 
        5, 4, 6, 7, 6, 4, 7, 7, 7, 7, 6, 7, 5, 7, 4, 7, 7, 4, 7, 7, 5, 5, 7, 
        7, 7, 5, 7, 4, 6, 6, 7, 6, 6, 7, 5, 5, 7, 7, 7, 5, 4, 4, 7, 6, 6, 6, 
        6, 7, 4, 4, 5, 7, 7, 6, 4, 4}; 
    data1 = Table[{n, Count[worldseries97, n]/Length[worldseries97]}, {n, 4, 
          7}];
x = 1/2; 
    data2 = {{4, (((1 - x))^4) + (x^4)}, {5, 
          4 (((1 - x))^4) x + 4 (1 - x) (x^4)}, {6, 
          10 (((1 - x))^4) (x^2) + 10 (((1 - x))^2) (x^4)}, {7, 
          20 (((1 - x))^4) (x^3) + 20 (((1 - x))^3) (x^4)}}; 
    gg = MultipleListPlot[data1, data2, 
        PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0]}, 
        PlotJoined -> True, Frame -> False, 
        PlotRange -> {0, 0.5}, Background -> GrayLevel[0.7]]; 
    gg2 = Show[{gg, {Graphics[{{RGBColor[0, 1, 0], 
                  Text["mathematics theory", {4.8, 0.3}]}, {RGBColor[1, 
                    0, 0], Text["World series", {5.5, 0.2}]}}]}}, 
        Background -> GrayLevel[0.7]]

4. Comparison between real data and the simulation by Mathematica.

We want to refute the assertion that the World Series has been fought between almost evenly matched teams. If we use the expression of mathematics, what we wanted to is to refute the statistical hypothesis that the World Series has been fought between almost evenly matched teams. Usually people use the theory of statistics to test the hypothesis, but in our case we had to compare probability distributions, and it was not easy thing to do for high school students.

Most of the students who participated in our research knew the notion of simulation, and hence the teacher chose to use computer simulation to test the statistical hypothesis.

Here we do the simulation 10000 times. In other words we repeat the history of the world series 10000 times.

4.1 Comparison between Table 2 and simulation.

If we compare Table 2 and Table 5, it is easy to find the fact that percentages  of 4 games and 7 games are a lot bigger in Table 2 than in Table 5. We have 19 years for the 4 games and 35 years for the 7 games.

First we did a computer simulation for the 97 years of the history of the World Series 10000 times using Mathematica, and we got only 11 cases that have more than 18 years of 4 games and more than 34 years of 7 games. 11 out of 10000 is 0.11%. In the tradition of statistics they think that 1% is a small chance, so 0.11% is very small, and hence it is rational to disregard the hypothesis that the World Series has been fought between almost evenly matched teams.

Mathematica simulation for the World Series from 1905 to 2005.

It took more than 2.5 minutes to do this simulation with my notebook computer. Perhaps it will take only less than 1.5 minute with new and faster computers.

Note that this is a simulation, and hence the new output is alwasy differnt from the previous output. In the following output the length of the list wshistory is 11, but in the next simulation the length may be longer or shorter. What is important is the fact that the number is relatively small compared to 10000.

Here wshistory is the list of the case that we have more than 18 years of 4 games and more than 34 years of 7 games

wshistory = {}; 
    Do[wsdata = {}; 
      Do[k = 0; h = 0; 
        While[k < 4 && h < 4, t = Random[Integer]; k = k + t; 
          h = h + (1 - t)]; 
        wsdata = Append[wsdata, k + h];, {n, 1, 97}]; 
If[
        Count[wsdata, 4] > 18 && Count[wsdata, 7] > 34, 
        wshistory = Append[wshistory, nn]], {nn, 1, 10000}]

wshistory

4.2 Comparison between Table 3 and simulation.

We do the same thing for Table 3. That is the 49 years of the history of the World Series, and we got only 4 cases that have more than 8 years of 4 games and more than 22 years of 7 games. 4 out of 10000 is 0.04%. 0.04% is extremely  rare, and hence it is rational to disregard the hypothesis.

Mathematica simulation for the World Series from 1956 to 2005.

It took more than 2 minutes to do this simulation with my notebook computer. Perhaps it will take only less than 1 minute with new and faster computers.

Note that this is a simulation, and hence the new output is alwasy differnt from the previous output. In the following output the length of the list wshistory is 4, but in the next simulation the length may be longer or shorter. What is important is the fact that the number is relatively small compared to 10000.

Here wshistory is the list of the case that we have more than 8 years of 4 games and more than 22 years of 7 games

wshistory = {}; 
      Do[wsdata = {}; 
        Do[k = 0; h = 0; 
          While[k < 4 && h < 4, t = Random[Integer]; k = k + t; 
            h = h + (1 - t)]; 
          wsdata = Append[wsdata, k + h];, {n, 1, 49}]; 
If[
          Count[wsdata, 4] > 8 && Count[wsdata, 7] > 22, 
          wshistory = Append[wshistory, nn]], {nn, 1, 10000}]

wshistory

4.3 Comparison between Table 4 and simulation.

We did the same thing for the data that professor P.J.Nahin used. This time we had 42 cases. 42 out of 10000 is 0.41%, it is still less than 1%. Therefore we have to disregard the hypothesis.

Mathematica simulation for the World Series from 1956 to 1999.

It took more than 2 minutes to do this simulation with my notebook computer. Perhaps it will take only less than 1 minute with new and faster computers.

Note that this is a simulation, and hence the new output is alwasy differnt from the previous output. In the following output the length of the list wshistory is 42, but in the next simulation the length may be longer or shorter.

wshistory = {}; 
      Do[wsdata = {}; 
        Do[k = 0; h = 0; 
          While[k < 4 && h < 4, t = Random[Integer]; k = k + t; 
            h = h + (1 - t)]; 
          wsdata = Append[wsdata, k + h];, {n, 1, 91}]; 
If[
          Count[wsdata, 4] > 16 && Count[wsdata, 7] > 32, 
          wshistory = Append[wshistory, nn]], {nn, 1, 10000}]

wshistory

{26, 520, 609, 723, 986, 1906, 1941, 2009, 2812, 3329, 3522, 4029, 
        4083, 4357, 4686, 4734, 4763, 4870, 4944, 5138, 5595, 5759, 6197, 
        6402, 6603, 6683, 6684, 6752, 6976, 6999, 7150, 7632, 7794, 7825, 
        8022, 8143, 8155, 9098, 9207, 9594, 9652, 9976} // Length

Remark. In this article we assumed that the probability of one team's win is constant and independent of the previous result of the game. Usually the probability depends on the result of the previous games.  We think that the calculated probabilities of the independent games are not different from the calculated probabilities of the non-independent games. One of the reason is the statistics of the Japan Series. It is reasonable to assume that the probabilities depends on the previous games in the Japan Series, but the mathematical theory based on the independence gave us a good model that explain the statistics of its history.

5.Probability distribution for the Series when we assume the home team advantage.

In the last section we refuted the assertion that the World Series has been fought between almost evenly matched teams.

Now we are going to try to explain the data in Tabel 2, 3 and 4 by making a new theory. One option is to assume that there have been more than two type of teams in the history of the World Series. For example you can assume that in a certain year team A has the probability p for winning against team B, whereas in another year team A has the probability of q, in another year the probability of r, etc. But if you study carefully 5, 6 and 7 games in Fig. 1 , you will notice that the probability for 6 games is always bigger than that of 7 games. Therefore this assumption does not explain Table 2, 3 and 4.

When we were studying the data in Table 2, 3 and 4, some of high school students proposed a new idea. They thought that home team advantage could explain the data. They presented this idea in 2002, and we talked about this at IMS2003. Shortly after that B. Stein and M. Heil [4]also presented the same idea.

Here we are going to make a theory that makes use of the concept of home team advantage.

In the rule of the World Series one team plays the first two games at home, and the last two games at home, if the series goes more than five games. We are going to formulate our theory under this rule.

Team A and B are going to play the game, and the 1st, 2nd and 3rd game is played in the stadium of the team A, and the 4th and 5th game in the stadium of the team B.

Team A and B are going to play the game, and the 1st, 2nd and 3rd game is played in the stadium of the team A, and the 4th and 5th game in the stadium of the team B.

Theorem 2. We suppose that the team A wins with the probability of p1 when it play agaist the team B in the stadium of the team A and the probability of p2 in the stadium of the team B.

(1) The probability that the World Series will require 4,5,6 or 7 is

1 - 2 p1 + (p1^2) - 2 p2 + 4 p1 p2 - 2 (p1^2) p2 + (p2^2) - 
      2 p1 (p2^2) + 2 (p1^2) (p2^2),

2 p1 - 4 (p1^2) + 2 (p1^3) + 2 p2 - 10 p1 p2 + 14 (p1^2) p2 - 
      4 (p1^3) p2 - 2 (p2^2) + 8 p1 (p2^2) - 
      8 (p1^2) (p2^2),

3 (p1^2) - 3 (p1^3) + 6 p1 p2 - 21 (p1^2) p2 + 16 (p1^3) p2 + 
      (p2^2) - 15 p1 (p2^2) + 42 (p1^2) (p2^2) - 30 (p1^3) (p2^2) - (p2^3) + 
      12 p1 (p2^3) - 30 (p1^2) (p2^3) + 20 (p1^3) (p2^3),

(p1^3) + 9 (p1^2) p2 - 12 (p1^3) p2 + 9 p1 (p2^2) - 
      36 (p1^2) (p2^2) + 30 (p1^3) (p2^2) + (p2^3) - 12 p1 (p2^3) + 
      30 (p1^2) (p2^3) - 20 (p1^3) (p2^3)

respectively.

(2) The average length of the games is

4 + 2 p1 + 2 (p1^2) - (p1^3) + 2 p2 + 2 p1 p2 - (p1^2) p2 - 
      8 (p1^3) p2 + 5 p1 (p2^2) - 32 (p1^2) (p2^2) + 30 (p1^3) (p2^2) + 
      (p2^3) - 12 p1 (p2^3) + 30 (p1^2) (p2^3) - 20 (p1^3) (p2^3)

Proof. We are going to use Mathematica to find the probabilities, because the calculation is very complicated.

Mathematica calculation for Theorem 2.

It is difficult to get the result of Theorem 2 by hand. This kind of calculation shows the power of Mathematica.

Here we use the function st[ ]. This function is the same as the function Strings[ ] defined in Pemmaraju and S. Skiena [3]. The reason is the same as we stated in the remark of Theorem 1.

We are going to calculate the case of 7 games.

Clear[a7, a6, a5, a4, p1, p2, c7, c6, c5, c4, x, k, t, tt7, tt6, tt5, 
        tt4, st];
st[l_List, 0]  :=  { {} };
st[l_List,  k_Integer]  :=  
        st[Union[l],  
            k]  /;  (Length[l]  =!=  Length[Union[l]]);
st[l_List, k_Integer]  :=  
        Distribute[Table[l,  {k}],  List];
a7 = 
        Select[st[{0, 1}, 7], 
          Count[#, 1] == 4 && 
              Part[#, 7] == 1 &];
b7 = 
        Select[st[{0, 1}, 7], 
          Count[#, 0] == 4 && 
              Part[#, 7] == 0 &];
c7 = Union[a7, b7];
t[x_, k_] := 
        If[k == 3 || k == 4 || k == 5, 
          x*p1 + (1 - x)*(1 - p1), 
          x*p2 + (1 - x)*(1 - p2)];
tt7[
          w_List] := (∏_(k = 1)^7) t[Part[w, k], k];

Clear[p1, p2];
game7 = Expand[Apply[Plus, Map[tt7, c7]]]

We are going to calculate the case of 6 games.

a6 = 
        Select[st[{0, 1}, 6], 
          Count[#, 1] == 4 && 
              Part[#, 6] == 1 &];
b6 = 
        Select[st[{0, 1}, 6], 
          Count[#, 0] == 4 && 
              Part[#, 6] == 0 &];
c6 = Union[a6, b6];
Clear[p1, p2];
t[x_, k_] := 
        If[k == 3 || k == 4 || k == 5, 
          x*p1 + (1 - x)*(1 - p1), 
          x*p2 + (1 - x)*(1 - p2)];
tt6[
          w_List] := (∏_(k = 1)^6) t[Part[w, k], k];

Clear[p1, p2];
game6 = Expand[Apply[Plus, Map[tt6, c6]]]

We are going to calculate the case of 5 games.

a5 = 
        Select[st[{0, 1}, 5], 
          Count[#, 1] == 4 && 
              Part[#, 5] == 1 &];
b5 = 
        Select[st[{0, 1}, 5], 
          Count[#, 0] == 4 && 
              Part[#, 5] == 0 &];
c5 = Union[a5, b5];
Clear[p1, p2];
t[x_, k_] := 
        If[k == 3 || k == 4 || k == 5, 
          x*p1 + (1 - x)*(1 - p1), 
          x*p2 + (1 - x)*(1 - p2)];
tt5[
          w_List] := (∏_(k = 1)^5) t[Part[w, k], k];

Clear[p1, p2];
game5 = Expand[Apply[Plus, Map[tt5, c5]]]

We are going to calculate the case of 4 games.

a4 = 
        Select[st[{0, 1}, 4], 
          Count[#, 1] == 4 && 
              Part[#, 4] == 1 &];
b4 = 
        Select[st[{0, 1}, 4], 
          Count[#, 0] == 4 && 
              Part[#, 4] == 0 &];
c4 = Union[a4, b4];
Clear[p1, p2];
t[x_, k_] := 
        If[k == 3 || k == 4 || k == 5, 
          x*p1 + (1 - x)*(1 - p1), 
          x*p2 + (1 - x)*(1 - p2)];
tt4[
          w_List] := (∏_(k = 1)^4) t[Part[w, k], k];

Clear[p1, p2];
game4 = Expand[Apply[Plus, Map[tt4, c4]]]

game7 + game6 + game5 + game4

expect = Expand[7*game7 + 6*game6 + 5*game5 + 4*game4]

6. Comparison between the theory and the data.

In the last subsection we proposed a new theory that makes use of the concept of home team advantage. But there is still a very important thing for us to do. It is to compare this theory and the empirical data from the history of the World Series. We know that the distribution of probabilities of the empirical data and that of probabilities derived from the theory is almost the same, but it remains to check all the home team games in the history.

The following example gives us a mathematical distribution that are near to the real data, but not very near. We found the conditions of Example 4 by try and error method.

We have to admit that we have to work harder to get a better theory!

Example 4. We make one hypothesis.  There two type of teams,and one is always strong and the other is strong in its home and weak otherwise.
    There are  70% of type (1) and 30% of type (2) teams in the history of the world series.
    (1) The team win with the probability of 0.9at its homeground games and 0.1 at away games.
    (2) The team win with the probability of 0.9at its homeground games and 0.8 at away games.

Then the probability that the World Series will require 4,5,6 or 7 is {0.16698,0.20472,0.202434,0.425866} and the average is 5.88719.

Mathematica calculation for the Example 4.

homead[p1_, 
          p2_] := {{1 - 2 p1 + (p1^2) - 2 p2 + 4 p1 p2 - 2 (p1^2) p2 + 
              (p2^2) - 2 p1 (p2^2) + 2 (p1^2) (p2^2), 
            2 p1 - 4 (p1^2) + 2 (p1^3) + 2 p2 - 10 p1 p2 + 
              14 (p1^2) p2 - 4 (p1^3) p2 - 2 (p2^2) + 8 p1 (p2^2) - 
              8 (p1^2) (p2^2), 
            3 (p1^2) - 3 (p1^3) + 6 p1 p2 - 21 (p1^2) p2 + 16 (p1^3) p2 + 
              (p2^2) - 15 p1 (p2^2) + 42 (p1^2) (p2^2) - 30 (p1^3) (p2^2) - 
              (p2^3) + 12 p1 (p2^3) - 30 (p1^2) (p2^3) + 20 (p1^3) (p2^3), 
            (p1^3) + 9 (p1^2) p2 - 12 (p1^3) p2 + 9 p1 (p2^2) - 
              36 (p1^2) (p2^2) + 30 (p1^3) (p2^2) + (p2^3) - 12 p1 (p2^3) + 
              30 (p1^2) (p2^3) - 20 (p1^3) (p2^3)}, 
          4 + 2 p1 + 2 (p1^2) - (p1^3) + 2 p2 + 2 p1 p2 - (p1^2) p2 - 
            8 (p1^3) p2 + 5 p1 (p2^2) - 32 (p1^2) (p2^2) + 
            30 (p1^3) (p2^2) + (p2^3) - 12 p1 (p2^3) + 30 (p1^2) (p2^3) - 
            20 (p1^3) (p2^3)};

homead[0.9, 0.1]*0.7 + homead[0.9, 0.8]*0.3

References

[1] P.J.Nahin, Duelling, Idiots and Other Probability Puzzlers, Princeton University Press, 2000.

[2] R. Miyadera, K. Miyabe, D. Kitajima, N. Fujii and K. Fujii,
How High School Students Could Present Original Math Research Using Mathematica, 
CHALLENGING THE BOUNDARIES OF SYMBOLIC COMPUTATION,
Proceedings of the Fifth International Mathematica Symposium,
Imperial College, 2003

[3] S. Pemmaraju and S. Skiena
Discrete Mathematics: Combinatorics and Graph Theory in Mathematica, Cambridge University Press,
2003.

[4]B. Stein and M. Heil,
Are 7-game World Series more common than expected? Inside Science News Service. American Institute of Physics. October 20, 2003. http://www.aip.org/isns/reports/2003/080.html