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Chapter 7: Problem 59

In the women's tennis tournament at Wimbledon, two finalists, \(\mathrm{A}\) and \(\mathrm{B}\), are competing for the title, which will be awarded to the first player to win two sets. In how many different ways can the match be completed?

Short Answer

Expert verified
There are 4 different ways in which the match can be completed: Player A wins two consecutive sets (AA), Player B wins two consecutive sets (BB), Player A wins the first and third sets (BAB), or Player B wins the first and third sets (ABB).

Step by step solution

01

Analyze Possible Events and Outcomes

There are three different possibilities on how the tournament can end: 1. Player A wins two consecutive sets. (AA) 2. Player B wins two consecutive sets. (BB) 3. The games are split, i.e., both players win one set each, and then either player A or player B wins the third deciding set. (ABB, BAB) Now, let's find out how many different ways the match can be completed in each scenario.
02

Calculate the Number of Possible Outcomes in Each Scenario

1. Player A wins two consecutive sets (AA). In this case, there is only 1 way, as player A wins both sets. 2. Player B wins two consecutive sets (BB). Similar to the previous case, there is only 1 way for this scenario. 3. The games are split, and either player A or player B wins the third set (ABB, BAB). First, let's focus on outcome ABB. Player A wins the first set, player B wins the second set, and player B wins the third set as well. There is 1 way to have this outcome. For outcome BAB, player B wins the first set, player A wins the second set, and player A wins the third set. Again, there is 1 way to have this outcome.
03

Sum Up the Number of Possible Outcomes to Find the Total Number of Outcomes

To find the total number of ways the match can be completed, we need to sum up the number of possible outcomes in each scenario. AA: 1 way BB: 1 way ABB: 1 way BAB: 1 way Total number of ways = 1 + 1 + 1 + 1 = 4 There are 4 different ways in which the match can be completed.

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Key Concepts

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

Probability
Probability allows us to determine the likelihood of different outcomes. In our tennis match example, each player's potential to win particular sets forms the basis of our outcomes. Think of the sequence Player A or B could win as individual events. We calculate each event's probability to understand the probability of the entire sequence. For instance, if the probability of each player winning a set is equal, we assume each outcome sequence (AA, BB, ABB, BAB) is equally likely. This means each sequence has a probability of 1 out of 4, or 0.25. The sum of the probabilities across all possible sequences must equal 1, indicating a certainty that one of those sequences will occur. Breaking down complex events into simpler discrete probabilities is a key aspect of mastering probability.
Set Theory
Set theory helps us describe collections of objects, here specifically sequences representing game outcomes. In our case, each sequence (AA, BB, ABB, BAB) can be seen as a distinct element of a set of all potential game outcomes. Consider this set as \(S = \{AA, BB, ABB, BAB\}\). Each element in \(S\) is a potential way the match might end. Set theory allows us to categorize and logically analyze all possibilities in a structured manner. This helps simplify the problem-solving process as it reframes the problem into a matter of identifying all possible sequences or outcomes and considering them collectively. With set theory, we can systematically determine exhaustive lists of events, confirm there are no omissions, and make calculations efficiently.
Sports Mathematics
Sports mathematics applies mathematical principles to sporting scenarios. In our tennis problem, it helps illustrate the logical approach to understanding the game's structure and outcomes. By modeling each match outcome as sequences of wins, we apply systematic counting to derive how many distinct results are possible. This aspect of sports mathematics blends probability and set theory, mapping sports events into mathematical notation. Such methods are crucial in analyzing tournament structures, optimizing strategies, and assessing performances. For practical understanding, players, coaches, and analysts use these insights to make informed decisions based on quantitative data.

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Most popular questions from this chapter

The number of cars entering a tunnel leading to an airport in a major city over a period of 200 peak hours was observed, and the following data were obtained: $$ \begin{array}{rc} \hline \begin{array}{l} \text { Number of } \\ \text { Cars, } x \end{array} & \begin{array}{c} \text { Frequency of } \\ \text { Occurrence } \end{array} \\ \hline 01000 & 15 \\ \hline \end{array} $$ a. Describe an appropriate sample space for this experiment. b. Find the empirical probability distribution for this experiment.

Let \(S=\left\\{s_{1}, s_{2}, s_{3}, s_{4}\right\\}\) be the sample space associated with an experiment having the probability distribution shown in the accompanying table. If \(A=\left\\{s_{1}, s_{2}\right\\}\) and \(B=\left\\{s_{1}, s_{3}\right\\}\), find a. \(P(A), P(B)\) b. \(P\left(A^{c}\right), P\left(B^{c}\right)\) c. \(P(A \cap B)\) d. \(P(A \cup B)\) $$ \begin{array}{lc} \hline \text { Outcome } & \text { Probability } \\ \hline s_{1} & \frac{1}{8} \\ \hline s_{2} & \frac{3}{8} \\ \hline s_{3} & \frac{1}{4} \\ \hline s_{4} & \frac{1}{4} \\ \hline \end{array} $$

The accompanying data were obtained from a survey of 1500 Americans who were asked: How safe are American-made consumer products? Determine the empirical probability distribution associated with these data. $$ \begin{array}{lccccc} \hline \text { Rating } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } \\ \hline \text { Respondents } & 285 & 915 & 225 & 30 & 45 \\ \hline \end{array} $$ A: Very safe B: Somewhat safe C: Not too safe D: Not safe at all E: Don't know

Let \(E\) and \(F\) be two events of an experiment with sample space \(S\). Suppose \(P(E)=.6, P(F)=.4\), and \(P(E \cap F)=\) .2. Compute: a. \(P(E \cup F)\) b. \(P\left(E^{c}\right)\) c. \(P\left(F^{c}\right)\) d. \(P\left(E^{c} \cap F\right)\)

The percentage of the general population that has each blood type is shown in the following table. Determine the probability distribution associated with these data. $$ \begin{array}{lcccc} \hline \text { Blood Type } & \text { A } & \text { B } & \text { AB } & \text { O } \\ \hline \text { Population, \% } & 41 & 12 & 3 & 44 \\ \hline \end{array} $$
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