Question12. The two-way table gives information on the performers in the New York Philharmonic Orchestra, categorized by section (type of instrument) and gender.
\begin{tabular}{c|c|c|c|c}
& \multicolumn{4}{c}{ Type of instrument } \\
& Strings & Woodwinds & Brass & Total \\
\cline { 2 - 6 } Gender & Male & 24 & 8 & 12 \\
\cline { 2 - 5 } & Female & 37 & 6 & 1 \\
\cline { 2 - 6 } & Total & 61 & 14 & 13 \\
\hline
\end{tabular}
You select one musician in this group at random. Which of the following statements is true about the events "Plays a woodwind" and "Male?
a) The events are mutually exclusive and independent.
b) The events are not mutually exclusive but they are independent.
c) The events are mutually exclusive but they are not independent.
d) The events are not mutually exclusive, nor are they independent.
e) The events are independent, but we do not have enough information to determine if they are mutually exclusive.
For questions 12 and 13 refer to the following: Mrs. Heart asked 100 randomly selected adult Americans if they thought that women should be allowed to go into combat situations. Here are the results, classified by the gender of the subject.
\begin{tabular}{c|c|c}
& \multicolumn{2}{c}{ Combat? } \\
\cline { 2 - 4 } Gender & Yes & No \\
\cline { 2 - 4 } & Male & 32 \\
\cline { 3 - 4 } & Female & 16 \\
& & 34
\end{tabular}
Studdy Solution
STEP 1
1. The two-way table provides data on the number of male and female performers in different instrument sections.
2. We are analyzing the relationship between the events "Plays a woodwind" and "Male."
3. The total number of musicians is 88.
STEP 2
1. Define mutually exclusive and independent events.
2. Determine if the events "Plays a woodwind" and "Male" are mutually exclusive.
3. Determine if the events "Plays a woodwind" and "Male" are independent.
STEP 3
Define mutually exclusive events: Two events are mutually exclusive if they cannot occur at the same time.
Define independent events: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
STEP 4
Check if the events "Plays a woodwind" and "Male" are mutually exclusive.
- From the table, there are 8 male woodwind players.
Since there are male woodwind players, the events are not mutually exclusive.
STEP 5
Check if the events "Plays a woodwind" and "Male" are independent.
Calculate the probability of "Plays a woodwind":
Calculate the probability of "Male":
Calculate the probability of "Plays a woodwind" and "Male":
Check for independence:
Since the probabilities are not equal, the events are not independent.
The correct statement is:
d) The events are not mutually exclusive, nor are they independent.
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