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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: \(\begin{array}{lr}\text { Bach } & 5 \% \\ \text { Beethoven } & 26 \% \\\ \text { Brahms } & 9 \% \\ \text { Dvorak } & 2 \% \\ \text { Mendelssohn } & 3 \% \\ \text { Mozart } & 21 \% \\ \text { Schubert } & 12 \% \\ \text { Schumann } & 7 \% \\ \text { Tchaikovsky } & 14 \% \\ \text { Wagner } & 1 \%\end{array}\) Suppose that one of these requests is to be selected at random. a. What is the probability that the request is for one of the three \(\mathrm{B}^{\prime}\) 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?

Step by step solution

01

Probability for the three 'B's

To find out the probability of a request for any of the three 'B's (Bach, Beethoven, or Brahms), we simply add the individual probabilities together. The probability of Bach is 5%, Beethoven 26%, and Brahms 9%. The combined probability is therefore \(5\% + 26\% + 9\% = 40\%\).

02

Probability not for the two 'S's

Calculating the probability of an event not happening involves subtracting the probability of the event from 100%. Schubert's probability is 12% and Schumann's is 7%. The combined probability of either Schubert or Schumann is \(12\% + 7\% = 19\%\). Therefore, the probability of a request not for either of these two composers is \(100\% - 19\% = 81\%\).

03

Probability for composers who wrote symphonies

Bach and Wagner did not write any symphonies. Therefore, we subtract their combined probability from 100%. Bach's probability is 5% and Wagner's is 1%. The combined probability of these two composers is \(5\% + 1\% = 6\%\). Therefore, the probability that the request is for a composer who did write symphonies is \(100\% - 6\% = 94\%\).

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Key Concepts

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

Classical Composers

When we think about classical music, certain composers often come to mind due to their remarkable contributions to the genre. In this exercise, we're talking about famous composers like Bach, Beethoven, Brahms, Dvorak, Mendelssohn, Mozart, Schubert, Schumann, Tchaikovsky, and Wagner. Each of these musicians has left a significant mark on music history. For example:

• Bach is known for his intricate fugues and profound choral works.
• Beethoven pushed the boundaries of classical music, connecting it to the Romantic era.
• Brahms, another Romantic composer, is famous for his rich symphonic textures.

These composers belong to varied times and styles, ranging from the Baroque period (Bach) to the Romantic period (Tchaikovsky and Wagner). Making requests for music from these composers reflects listeners' diverse tastes in classical music.

Statistical Calculation

Statistical calculation helps quantify and make sense of random data scenarios. Upon encountering probability percentages, like those given for classical composers, statistics allow us to reassemble and interpret individual data points for deeper insights. In this context, every composer's probability of being requested suggests the frequency of their music's popularity among listeners.
For our exercise,

• the probability for each composer is given as a percentage.
• These percentages must add up to 100%, which covers all possible outcomes for the requests.
• Calculating the probability for multiple events, like the three 'B's (Bach, Beethoven, Brahms), involves simple arithmetic—adding these relevant percentages together.
• This is fundamental when predicting the likelihood of different outcomes on a specific night of requests at the radio station.

Statistical calculations are critical in various fields, from market analysis to music requests, as they translate raw data into understandable outcomes.

Probability Calculation

Probability calculation is essential when we want to understand the likelihood of specific events occurring from a given set of possibilities. In the context of our classical music requests example, probability helps us determine which composer might be chosen.
To calculate the probability, we add up the percentages of the desired outcomes:

• The probability of choosing one of the three 'B's (Bach, Beethoven, Brahms) is calculated by summing their individual probabilities: Bach (5%), Beethoven (26%), Brahms (9%). This gives us a total of 40%.
• For probabilities involving non-selection, we subtract the sum of undesired outcomes from 100%. For example, the probability of not picking Schubert or Schumann (12% + 7% = 19%) is 81% when subtracted from 100%.
• Calculating for composers who wrote symphonies involves removing those who did not from our total. For Bach and Wagner, this results in subtracting 6% (Bach and Wagner) from 100% to yield a 94% likelihood.

Probability calculations make it easier to visualize potential outcomes and manage expectations in situations involving chance.