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Chapter 6: Problem 31

A radio station that plays classical music has a "by request" program each Saturday evening. The percentages of requests for composers on a particular night are as follows: \(\begin{array}{lr}\text { Bach } & 5 \% \\ \text { Mozart } & 21 \% \\ \text { Beethoven } & 26 \% \\ \text { Schubert } & 12 \% \\ \text { Brahms } & 9 \% \\ \text { Schumann } & 7 \% \\ \text { Dvorak } & 2 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Wagner } & 1 \%\end{array}\) Suppose that one of these requests is randomly selected. a. What is the probability that the request is for one of the three B's? b. What is the probability that the request is not for one of the two S's? c. Neither Bach nor Wagner wrote any symphonies. What is the probability that the request is for a composer who wrote at least one symphony?

Short Answer

Expert verified
a. The probability that the request is for one of the three B's is 40%. b. The probability that the request is not for one of the two S's is 81%. c. The probability that the request is for a composer who wrote at least one symphony is 94%.

Step by step solution

01

Calculation of the probability for the three B's

The three composers mentioned here are Bach, Beethoven, and Brahms. Hence, the probability that a request is for one of these is simply the sum of their individual percentages: \[5\% + 26\% + 9\%\]
02

Calculation of the probability for not one of the two S's

The two composers mentioned here are Schubert and Schumann. To find the probability that the request is not for one of these, subtract their individual percentages from 100%: \[100\% - 12\% - 7\%\]
03

Calculation of the probability for a composer who wrote at least one symphony

The composers who did not write any symphonies are Bach and Wagner. Hence, the probability that the request is for a composer who wrote at least one symphony is 100% minus the sum of their percentages: \[100\% - 5\% - 1\%\]

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Key Concepts

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

Classical Composers
Classical composers are the brilliant minds behind the masterpieces of music from centuries past. Think of them as the "rock stars" of their time, crafting symphonies that still resonate with audiences today. Some of the famous names you may encounter include Bach, whose intricate compositions are mathematical in precision, and Beethoven, known for his powerful and emotional symphonies. Classical music itself often revolves around set forms like sonatas, symphonies, and concertos that composers used to explore themes and variations. Understanding these composers helps us appreciate not only their historical context but also the influence they continue to exert on modern music. Each composer brought a unique style to their music, contributing vastly to the cultural treasures we enjoy today.
Music Requests
Music requests are a fun way for listeners to engage with radio stations and hear their favorite classical pieces. In the context of a classical music station, listeners may request pieces from particular composers, allowing them to interact with the music they love. The concept of a "by request" program means that listeners have a hand in shaping the playlist for a given segment, typically by choosing from a list of well-known composers. This shared experience not only strengthens listener participation but also reflects the enduring popularity of certain pieces or composers. It is not uncommon to see recurring requests for composers like Bach, Beethoven, and Mozart, whose works have become staples in classical music programming.
Symphonies
Symphonies are one of the most robust and dynamic forms of classical music. Essentially, a symphony is a large, complex orchestral piece usually composed of multiple movements. These movements vary in mood and tempo, often starting with a fast-paced and commanding opening, followed by more measured sections. Famous for their grandeur and emotional depth, symphonies can convey anything from thunderous drama to serene beauty. Not all classical composers wrote symphonies; famous names like Wagner and Bach are known for other forms of compositions. However, symphonies by composers such as Beethoven have left a monumental impact on music. They stand as the ultimate testaments to a composer's ability to develop and weave intricate musical ideas into cohesive works.
Percentage Calculations
Percentage calculations are a vital tool in various fields, including data analysis and probability. When applied to music requests, percentages can help us understand the proportion of requests for each composer. To calculate the probability of an event, such as the likelihood of a particular composer being requested, you simply treat the percentage as a part of a whole.
For example, if Beethoven is requested 26% of the time, that implies a 26 in 100 chance for a random selection to be one of his pieces. It's like rolling a specialized die that has Beethoven's name on 26 out of 100 faces!
Understanding percentages allows us to quickly gauge probabilities and make informed predictions based on historical data. It's a skillset that blends mathematics with intuitive reasoning, invaluable for both academic and real-world scenarios.

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Most popular questions from this chapter

\(6.23\) A medical research team wishes to evaluate two different treatments for a disease. Subjects are selected two at a time, and then one of the pair is assigned to each of the two treatments. The treatments are applied, and each is either a success (S) or a failure (F). The researchers keep track of the total number of successes for each treatment. They plan to continue the chance experiment until the number of successes for one treatment exceeds the number of successes for the other treatment by \(2 .\) For example, they might observe the results in the table on the next page. The chance experiment would stop after the sixth pair, because Treatment 1 has two more successes than Treatment \(2 .\) Treatment 1 and success for Treatment 2 . Continue to select pairs, keeping track of the cumulative number of successes for each treatment. Stop the trial as soon as the number of successes for one treatment exceeds that for the other by 2 . This would complete one trial. Now repeat this whole process until you have results for at least 20 trials [more is better]. Finally, use the simulation results to estimate the desired probabilities.) a. What is the probability that more than 5 pairs must be treated before a conclusion can be reached? (Hint: \(P\) (more than 5\()=1-P(5\) or fewer \() .)\) b. What is the probability that the researchers will incorrectly conclude that Treatment 2 is the better treatment?

Researchers at UCLA were interested in whether working mothers were more likely to suffer workplace injuries than women without children. They studied 1400 working women, and a summary of their findings was reported in the San Luis Obispo Telegram-Tribune (February 28,1995 ). The information in the following table is consistent with summary values reported in the article: $$\begin{array}{l|cccc} & & & \text { Children, } & \\ & \text { No } & \text { Children } & \text { but None } & \\ & \text { Children } & \text { Under 6 } & \text { Under 6 } & \text { Total } \\\ \hline \text { Injured on } & & & & \\ \text { the Job in } & & & & \\ 1989 & 32 & 68 & 56 & \mathbf{1 5 6} \\ \text { Not Injured } & & & & \\ \text { on the Job } & & & & \\ \text { in 1989 } & 368 & 232 & 644 & \mathbf{1 2 4 4} \\ \text { Total } & \mathbf{4 0 0} & \mathbf{3 0 0} & \mathbf{7 0 0} & \mathbf{1 4 0 0} \end{array}$$ The researchers drew the following conclusion: Women with children younger than age 6 are much more likely to be injured on the job than childless women or mothers with older children. Provide a justification for the researchers' conclusion. Use the information in the table to calculate estimates of any probabilities that are relevant to your justification.

Three friends \((\mathrm{A}, \mathrm{B}\), and \(\mathrm{C})\) will participate in a round-robin tournament in which each one plays both of the others. Suppose that \(P(\mathrm{~A}\) beats \(\mathrm{B})=.7, P(\mathrm{~A}\) beats \(\mathrm{C})\) \(=.8\), and \(P(\) B beats \(C\) ) \(=.6\) and that the outcomes of the three matches are independent of one another. a. What is the probability that \(\mathrm{A}\) wins both her matches and that B beats \(C\) ? b. What is the probability that A wins both her matches? c. What is the probability that A loses both her matches? d. What is the probability that each person wins one match? (Hint: There are two different ways for this to happen. Calculate the probability of each separately, and then add. )

Approximately \(30 \%\) of the calls to an airline reservation phone line result in a reservation being made. a. Suppose that an operator handles 10 calls. What is the probability that none of the 10 calls results in a reservation? b. What assumption did you make in order to calculate the probability in Part (a)? c. What is the probability that at least one call results in a reservation being made?

A Gallup Poll conducted in November 2002 examined how people perceived the risks associated with smoking. The following table summarizes data on smoking status and perceived risk of smoking that is consistent with summary quantities published by Gallup: $$\begin{array}{l|cccc} \hline \text { Smoking } & \begin{array}{c} \text { Very } \\ \text { Status } \end{array} & \begin{array}{c} \text { Some- } \\ \text { what } \\ \text { Harmful } \end{array} & \begin{array}{c} \text { Not } \\ \text { Too } \\ \text { Harmful } \end{array} & \begin{array}{c} \text { Not } \\ \text { at All } \\ \text { Harmful } \end{array} & \text { Harmful } \\ \hline \begin{array}{l} \text { Current } \\ \text { Smoker } \\ \text { Former } \end{array} & 60 & 30 & 5 & 1 \\ \text { Smoker } & 78 & 16 & 3 & 2 \\ \begin{array}{l} \text { Never } \\ \text { Smoked } \end{array} & 86 & 10 & 2 & 1 \\ \hline \end{array}$$ Assume that it is reasonable to consider these data representative of the U.S. adult population. Consider the following conclusion: Current smokers are less likely to view smoking as very harmful than either former smokers or those who have never smoked. Provide a justification for this conclusion. Use the information in the table to calculate estimates of any probabilities that are relevant to your justification.
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