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Professional basketball is now a reality for women basketball players in the United States. There are two conferences in the WNBA, each with six teams, as shown in the following table. \(^{3}\) $$ \begin{array}{ll} \hline \text { Western Conference } & \text { Eastern Conference } \\ \hline \text { Minnesota Lynx } & \text { Atlanta Dream } \\ \text { Phoenix Mercury } & \text { Indiana Fever } \\ \text { Dallas Wings } & \text { New York Liberty } \\ \text { Los Angeles Sparks } & \text { Washington Mystics } \\ \text { Seattle Storm } & \text { Connecticut Sun } \\ \text { San Antonio Stars } & \text { Chicago Sky } \end{array} $$ Two teams, one from each conference, are randomly selected to play an exhibition game. a. How many pairs of teams can be chosen? b. What is the probability that the two teams are Los Angeles and New York? c. What is the probability that the Western Conference team is not from California?

Step by step solution

01

Part a: Number of Pairs of Teams

To find the number of possible pairs of teams for an exhibition game, notice that each team in the Western Conference can be paired with each team in the Eastern Conference, making a total number of pairs equal to the product of the number of teams in each conference. There are 6 teams in each conference, so there are \(6\times6 = 36\) possible pairs of teams that can be chosen.

02

Part b: Probability of Los Angeles and New York Playing

To find the probability that the two teams playing are Los Angeles and New York, we will use the definition of probability: \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). There is only one favorable outcome, the pair of Los Angeles Sparks and New York Liberty out of 36 possible pairs of teams. So, the probability is \(\frac{1}{36}\).

03

Part c: Probability of the Western Conference Team Not Being from California

To find the probability that the Western Conference team is not from California, we need to find another way to express the desired probability. Note that the number of teams in the Western Conference not from California is 4 (Minnesota, Phoenix, Dallas, and San Antonio). Each of these 4 teams can be paired with any of the 6 teams in the Eastern Conference, yielding 24 favorable outcomes. The total number of outcomes is still 36, so the desired probability is \(\frac{24}{36} = \frac{2}{3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics

Combinatorics involves the study of counting and arrangement of events or objects. In sports, combinatorial techniques help determine the number of different possible matchups between teams. Let's consider the WNBA, where we have the Western and Eastern Conferences, each having six teams. When creating exhibition game pairings, each Western team can match with any Eastern team. This shows the essence of combinatorics in sports, as calculations are done in combinations and products. For instance, with six teams in each conference:

• Number of Western combinations: 6
• Number of Eastern combinations: 6
• Total possible matchups: 6 × 6 = 36

The straightforward clinic of multiplication allows us to see the different pair combinations, helping leagues plan exhibitions or schedules effectively.

Conditional Probability

Conditional probability evaluates the probability of an event occurring, given that another event happens. It can be particularly useful in sports predictions. For instance, if we're interested in knowing the likelihood of the Los Angeles Sparks and New York Liberty playing together, this is about analyzing a specific case among all possible cases. To use conditional probability:

• Identify favorable outcomes: There is only one - Los Angeles and New York.
• Divide by total outcomes: Total of 36 matchups possible.
• Probability: \( \frac{1}{36} \)

Thus, the probability of these two teams playing together in an exhibition match is \( \frac{1}{36} \), illustrating how conditional probability narrows down possibilities to specific scenarios.

WNBA

The Women's National Basketball Association (WNBA) has grown significantly, providing professional opportunities for women in the sport. Within the league, both Western and Eastern Conferences compete, each conference comprising six vibrant teams. These teams showcase some of the best basketball talents from across the globe, making the league a critical element in the promotion and development of women's basketball.
The WNBA utilizes complex scheduling and statistical methods, like those in questions of probability and combinatorics, to ensure the smooth delivery of matches. The ability to calculate pairings and match probabilities is crucial in arranging games and tournaments that are engaging and fair. Ultimately, this structured approach not only serves as a platform for player development but also fosters fan engagement, fueling the growth of basketball among diverse audiences.