Question

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 table below. \(^{3}\) 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?

Short answer

Answer: There can be 36 different pairs of teams chosen for the exhibition game. Question: What is the probability that the two teams are Los Angeles and New York? Answer: The probability of Los Angeles and New York being chosen as the teams for the exhibition game is 1/36. Question: What is the probability that the Western Conference team is not from California? Answer: The probability that the Western Conference team is not from California is 5/6.

Step-by-step solution

Find the Total Number of Possible Pairs of Teams

We have two conferences, each with six teams. Since one team is randomly chosen from each conference, we can find the total number of pairs by multiplying the number of teams in each conference: Total pairs = (Number of teams in Conference 1) * (Number of teams in Conference 2) Total pairs = 6 teams * 6 teams Total pairs = 36 pairs #a. How many pairs of teams can be chosen?# There can be 36 different pairs of teams chosen for the exhibition game.

Probability of Los Angeles and New York being chosen as the teams

To find the probability of Los Angeles and New York being the two teams chosen, first, we need to know that there is only 1 favorable outcome (Los Angeles and New York playing each other). To find the probability, we will use the ratio between the number of favorable outcomes and the total number of outcomes: Probability (Los Angeles and New York) = (Number of favorable outcomes) / (Total number of outcomes) Probability (Los Angeles and New York) = 1 / 36 #b. What is the probability that the two teams are Los Angeles and New York?# The probability of Los Angeles and New York being chosen as the teams for the exhibition game is 1/36.

Probability of the Western Conference team not being from California

Since Los Angeles is the only team from California in the Western Conference, we are looking for 5 teams out of 6 for the Western Conference team choice. We have 5 favorable outcomes for the Western Conference team, with 6 possibilities in the Eastern Conference. To find the probability, we will use the ratio between the number of favorable outcomes and the total number of outcomes: Probability (Western Conference team not from California) = (Number of favorable outcomes) / (Total number of outcomes) Probability (Western Conference team not from California) = (5 Western Conference choices * 6 Eastern Conference choices) / (6 Western Conference choices * 6 Eastern Conference choices) Probability (Western Conference team not from California) = 30 / 36 = 5/6 #c. What is the probability that the Western Conference team is not from California?# The probability that the Western Conference team is not from California is 5/6.

Key concepts

Combinatorics

Combinatorics deals with counting, arranging, and selecting objects in a set. It's a branch of mathematics that focuses on the number of ways objects can be chosen from a larger set. In the context of the WNBA scenario, combinatorics helps determine the number of possible pairs of teams for an exhibition game.

To find the total number of pairs, we multiply the number of teams in each conference. With two conferences, each containing six teams, the number of ways to select one team from each conference is simply:

• Total pairs = \(6 \times 6 = 36\)

This result means there are 36 different team pairings possible when choosing one team from each conference. By using basic combinatorial principles, we quickly determine the ways to form these pairs without listing every possible combination.

Women's National Basketball Association (WNBA)

The Women's National Basketball Association (WNBA) is a professional basketball league in the United States. It represents professional female basketball players and is divided into two main conferences. Each conference contains six teams, providing a competitive platform for female athletes.

The WNBA, established as a counterpart to the NBA, offers opportunities for talented women to play basketball at a professional level. An exhibition game, as described in the problem, serves as a showcase match that may not affect the regular-season standings but offers an exciting matchup for fans. By random selection of teams from each conference, the exhibition highlights the diversity of teams in the league.

Understanding how these selections work is crucial for those interested in sports management or probability within sports, as it reflects the real-world scenarios teams might encounter during event planning.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring given that another event has already occurred. It is an essential concept in probability that helps to make predictions in real-world situations, such as those found in the WNBA team selections.

In the problem, consider the probability that the Western Conference team is not from California. The condition here is that one team has to come from the Western Conference. There are 5 non-California teams in this conference. Thus:

• Favorable outcomes from the Western Conference = 5
• Total possible team pairings based on conference requirements = \(36\)

The probability calculation uses the ratio of favorable outcomes over the total possibilities:

• Probability = \( \frac{5 \times 6}{6 \times 6} = \frac{30}{36} = \frac{5}{6} \)

This calculation shows that given the team is from the Western Conference, there is a high chance that it is not a California team. Understanding this helps in making more informed choices and predictions in statistical and sports-related analyses.