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Chapter 2: Problem 1

Four universities- \(1,2,3\), and 4 -are participating in a holiday basketball tournament. In the first round, 1 will play 2 and 3 will play 4 . Then the two winners will play for the championship, and the two losers will also play. One possible outcome can be denoted by 1324 (1 beats 2 and 3 beats 4 in first- round games, and then 1 beats 3 and 2 beats 4 ). a. List all outcomes in \(\mathcal{S}\). b. Let \(A\) denote the event that 1 wins the tournament. List outcomes in \(A\). c. Let \(B\) denote the event that 2 gets into the championship game. List outcomes in \(B\). d. What are the outcomes in \(A \cup B\) and in \(A \cap B\) ? What are the outcomes in \(A^{\prime}\) ?

Short Answer

Expert verified
a. 16 outcomes; b. 1324, 1342, 1423, 1432; c. 2314, 2341; d. \(A \cup B\): all listed; \(A \cap B\): none; \(A'\): all not in b.

Step by step solution

01

Understanding the Possible Outcomes

The tournament involves two rounds. In the first round: 1 vs 2 and 3 vs 4. In the second round: the two winners play against each other, and the two losers play against each other. An outcome is represented by the sequence of game results, such as 1324.
02

Listing All Possible Outcomes (Part a)

Each first-round game has 2 possible outcomes (either team can win). Hence, there are 2 outcomes for the first match and 2 outcomes for the second match. The second round also has 2 possible outcomes for each match. Therefore, the total number of outcomes is \[ 2 \times 2 \times 2 \times 2 = 16 \]. List of outcomes: 1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231.
03

Identifying Outcomes where University 1 Wins (Part b)

For university 1 to win, they must win both the first and the second round games. The possible outcomes are: 1324, 1342, 1423, 1432.
04

Identifying Outcomes where University 2 Reaches the Championship Game (Part c)

For university 2 to reach the championship, they must win in the first round. Possible outcomes are: 2314, 2341, 2413, 2431, 2314, 2341.
05

Finding the Union and Intersection of Events A and B (Part d)

- **Outcomes in \(A \cup B\):** These include all outcomes where either event \(A\) or event \(B\) occurs or both. Hence, it's the combination of both outcomes: 1324, 1342, 1423, 1432, 2314, 2341, 2413, 2431.- **Outcomes in \(A \cap B\):** These are the outcomes where both events occur, i.e., university 1 wins and university 2 reaches the championship. No such outcomes exist, so it's an empty set \(\emptyset\).
06

Finding Outcomes Not in Event A (Part d)

- **Outcomes in \(A^{\prime}\):** These are outcomes from \(\mathcal{S}\) that are not in \(A\): 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 4123, 4132, 4213, 4231.

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Key Concepts

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

Outcomes
In a tournament setting, an "outcome" represents a complete sequence of events that describe how games are resolved. For our basketball tournament example, outcomes are designated by sequences like "1324", indicating:
  • 1 beats 2
  • 3 beats 4
  • 1 beats 3 in the championship
  • 2 beats 4 among the losers
Every possible sequence from the start to the end of the games counts as a distinct outcome. For any single event where teams compete, each matchup has two possibilities - one team wins, or the other team wins. Hence, each round, provided it contains independent matchups, multiplies the possible outcomes.
Events
An event, in probability terms, is defined as a set or a subset of outcomes from the sample space of all potential outcomes (\(\mathcal{S}\)). It describes a particular condition or scenario of interest. In our scenario, if we consider the event \(A\), where University 1 wins the tournament, it includes all outcomes wherein University 1 successfully wins both rounds. Similarly, event \(B\) specifically refers to outcomes where University 2 makes it to the championship game. These events allow us to focus on scenarios of interest among all possible outcomes.
Intersection and Union
When considering events, two essential concepts are intersection and union. **Intersection** (\(A \cap B\)) refers to outcomes that are common to both events \(A\) and \(B\). In our tournament scenario, this would mean outcomes where University 1 wins the tournament, and University 2 reaches the championship game. In this case, no such outcomes exist, hence, \(A \cap B = \emptyset\).
The **Union** (\(A \cup B\)) combines all outcomes from either event \(A\), \(B\), or both. This includes any scenario where either University 1 wins or University 2 makes it to the championship. It encompasses a broader set of outcomes than the intersection.
Complement of an Event
The complement of an event \(A\), denoted as \(A'\), includes all outcomes in the sample space \(\mathcal{S}\) that are not in \(A\). Put simply, \(A'\) captures what happens outside the specific scenario where University 1 wins. This perspective allows students to consider "non-occurrence" of event \(A\) and helps in calculating probabilities indirectly by subtracting the probability of \(A\) from 1. In the context of our problem, listing outcomes in \(A'\) means characterizing every possible game result where University 1 does not emerge as the winner.

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