DescriptionThe World Series is the annual championship series of Major League Baseball (MLB) to determine that season’s MLB champion. It dates back to 1903, and there has been a World Series every year since except for 1904 (boycott) and 1994 (strike). The format of the World Series is that the first team to win four games of the series wins the World Series. This format is called “best-of-seven,” since it takes a maximum of seven games for one team to win four games. There were four World Series however that were “best-of-nine,” requiring a team to win five games to win the World Series: 1903, 1919, 1920, and 1921 (in 1919, MLB decided to experiment with a best-of-nine format, partly to increase baseball’s popularity and partly to generate more revenue, but it reverted back to best-of-seven permanently in 1922). The most successful team are the New York Yankees with 27 World Series Titles (40 World Series appearances), while the Seattle Mariners are the only MLB team without a World Series appearance.

The focus of this analytic is on five high school juniors’ reasoning as they calculate the probabilities that the World Series will be won in four, five, six, or seven games. According to Researcher Kiczek, in the PUP Math “World Series” interview, a goal was “to see if [the students] could solve a particular probability problem without having been taught how to do it, without any formal rules or notation or anything being imposed.” Researcher Maher presented the World Series Problem to the group of five high-school juniors: Michael, Jeff, Ankur, Romina, and Brian. Romina conjectured that the probability the series would end in four games was 1/16. This did not take into account the combined probabilities for either team winning in four games. In Event 2, the students work on calculating possibilities for five games. Their approach was to count the number of possible outcomes for five games by listing the possible outcomes as strings of A’s and B’s (for Team A and Team B). They notice a need to restrict the sample space to the outcomes in which the last game is won by the winning team. After the students determine the number of outcomes for five games, they focus on computing the exact probability. According to Researcher Kiczek: “for a five-game series, it was a little bit more complicated, and they realized that they got 8 different strings. But when they tried to figure out what the probability of that was, they knew it was 8 over something, and it was the ‘over something’ part that they had a little trouble with.” The students not only determine what that “over something” should be in Event 3, they also determine the denominators for all of the cases, using the combinatorics rule of 2^n that they had discovered in a previous problem-solving session. They continue with the case of winning in the 6th game in Event 4. They determine the numerator for computing the probability for winning in six games. They notice that they only needed to list outcomes for a single team winning the World Series, and then double that result to find the probability of either team winning, referring to the other team’s outcome as “opposites.” For a win in Game 6, they found 10 outcomes for one team winning the World Series, which led to the numerator of 20. They determine the final probability for seven games to be 40/128 in Event 5. In doing so, Ankur and Jeff apply a fundamental probability concept that all of the probabilities sum to 1. While they verify all of the probabilities in Event 6, Ankur offers a generalization that the probability of success can be expressed as the ratio of the number of favorable outcomes to the number of total outcomes. Notice in Event 6 that Romina realized that she failed earlier to account for either team winning in calculating the probability of four games. The students present their solutions for the probability of four games in Event 7 to Researchers Maher and Alston, and in Event 8, offer the probabilities of a World Series win in five, six, and seven games. In Event 9, a week later, the students present their solution to Researcher Speiser for the probability of a win in four and five games. In Event 10, the students share their solution of the probability of winning for a World Series that ends in six and seven games.

While working on this task, notice that the students represented their outcomes using strings of A’s and B’s, and later with Researcher Speiser (Events 9 and 10), strings of H’s and A’s to represent the home team and the away team, respectively. They then counted up the favorable outcomes for each case, and they knew that this number represents the numerator of the probability, while the total possible outcomes – which they eventually figured out is calculated by 2^n for a series of n games – represents the denominator of the probability. Notice, also, that Michael pointed out a relationship between the numerators of the probabilities and Pascal’s Triangle, specifically the diagonal of Pascal’s Triangle consisting of the fourth entries of Rows 3-6. These students may not have discovered the general formula P(x) = 2*C(x – 1, x – w)*(1/2)^x, where x is the number of games in the series and w is the number of wins that is needed to win the World Series (in this case, w = 4), but their collaboration and intuitions throughout their reasoning in this challenging task were very impressive, especially considering their basic knowledge of probability.

Task statement: In a “world series” two teams play each other in at least four and at most 7 games. The first team to win four games is the winner of the “world series.” Assuming that both teams are equally matched, what is the probability that a “world series” will be won:

a) In 4 games?

b) In 5 games?

c) In 6 games?

d) In 7 games?

Video references:

1. B37, World Series problem (student view), Grade 11, January 22, 1999, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T33R0R0Q

2. B39, Revisiting the World Series problem (student view), Grade 11, January 29, 1999, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3S75DGS

3. PUP Math World series [video]. Retrieved from https://doi.org/doi:10.7282/T3CV4H0V

Non-video reference:

Kiczek, Regina Dockweiler. (2000). Tracing the Development of Probabilistic Thinking: Profiles from a Longitudinal Study. Doctoral dissertation, Rutgers University, NJ.

The focus of this analytic is on five high school juniors’ reasoning as they calculate the probabilities that the World Series will be won in four, five, six, or seven games. According to Researcher Kiczek, in the PUP Math “World Series” interview, a goal was “to see if [the students] could solve a particular probability problem without having been taught how to do it, without any formal rules or notation or anything being imposed.” Researcher Maher presented the World Series Problem to the group of five high-school juniors: Michael, Jeff, Ankur, Romina, and Brian. Romina conjectured that the probability the series would end in four games was 1/16. This did not take into account the combined probabilities for either team winning in four games. In Event 2, the students work on calculating possibilities for five games. Their approach was to count the number of possible outcomes for five games by listing the possible outcomes as strings of A’s and B’s (for Team A and Team B). They notice a need to restrict the sample space to the outcomes in which the last game is won by the winning team. After the students determine the number of outcomes for five games, they focus on computing the exact probability. According to Researcher Kiczek: “for a five-game series, it was a little bit more complicated, and they realized that they got 8 different strings. But when they tried to figure out what the probability of that was, they knew it was 8 over something, and it was the ‘over something’ part that they had a little trouble with.” The students not only determine what that “over something” should be in Event 3, they also determine the denominators for all of the cases, using the combinatorics rule of 2^n that they had discovered in a previous problem-solving session. They continue with the case of winning in the 6th game in Event 4. They determine the numerator for computing the probability for winning in six games. They notice that they only needed to list outcomes for a single team winning the World Series, and then double that result to find the probability of either team winning, referring to the other team’s outcome as “opposites.” For a win in Game 6, they found 10 outcomes for one team winning the World Series, which led to the numerator of 20. They determine the final probability for seven games to be 40/128 in Event 5. In doing so, Ankur and Jeff apply a fundamental probability concept that all of the probabilities sum to 1. While they verify all of the probabilities in Event 6, Ankur offers a generalization that the probability of success can be expressed as the ratio of the number of favorable outcomes to the number of total outcomes. Notice in Event 6 that Romina realized that she failed earlier to account for either team winning in calculating the probability of four games. The students present their solutions for the probability of four games in Event 7 to Researchers Maher and Alston, and in Event 8, offer the probabilities of a World Series win in five, six, and seven games. In Event 9, a week later, the students present their solution to Researcher Speiser for the probability of a win in four and five games. In Event 10, the students share their solution of the probability of winning for a World Series that ends in six and seven games.

While working on this task, notice that the students represented their outcomes using strings of A’s and B’s, and later with Researcher Speiser (Events 9 and 10), strings of H’s and A’s to represent the home team and the away team, respectively. They then counted up the favorable outcomes for each case, and they knew that this number represents the numerator of the probability, while the total possible outcomes – which they eventually figured out is calculated by 2^n for a series of n games – represents the denominator of the probability. Notice, also, that Michael pointed out a relationship between the numerators of the probabilities and Pascal’s Triangle, specifically the diagonal of Pascal’s Triangle consisting of the fourth entries of Rows 3-6. These students may not have discovered the general formula P(x) = 2*C(x – 1, x – w)*(1/2)^x, where x is the number of games in the series and w is the number of wins that is needed to win the World Series (in this case, w = 4), but their collaboration and intuitions throughout their reasoning in this challenging task were very impressive, especially considering their basic knowledge of probability.

Task statement: In a “world series” two teams play each other in at least four and at most 7 games. The first team to win four games is the winner of the “world series.” Assuming that both teams are equally matched, what is the probability that a “world series” will be won:

a) In 4 games?

b) In 5 games?

c) In 6 games?

d) In 7 games?

Video references:

1. B37, World Series problem (student view), Grade 11, January 22, 1999, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T33R0R0Q

2. B39, Revisiting the World Series problem (student view), Grade 11, January 29, 1999, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T3S75DGS

3. PUP Math World series [video]. Retrieved from https://doi.org/doi:10.7282/T3CV4H0V

Non-video reference:

Kiczek, Regina Dockweiler. (2000). Tracing the Development of Probabilistic Thinking: Profiles from a Longitudinal Study. Doctoral dissertation, Rutgers University, NJ.

Created on2022-07-06T17:23:39-0400

Published on2022-07-29T10:44:49-0400

Persistent URLhttps://doi.org/doi:10.7282/t3-5gza-9a67