Answer :
To determine which of the given probabilities are examples of independent events, let's analyze the given data step-by-step and calculate each of these probabilities.
### Step-by-Step Calculation:
#### Step 1: Total counts of preferences
First, calculate the total number of teachers and librarians:
- Teachers:
- Movies: 3
- Books: 4
- Plays: 2
- Total Teachers = 3 + 4 + 2 = 9
- Librarians:
- Movies: 12
- Books: 9
- Plays: 15
- Total Librarians = 12 + 9 + 15 = 36
Next, calculate the overall totals for each activity:
- Movies:
- Total Movies = 3 (teachers) + 12 (librarians) = 15
- Books:
- Total Books = 4 (teachers) + 9 (librarians) = 13
- Plays:
- Total Plays = 2 (teachers) + 15 (librarians) = 17
- Total People (Teachers + Librarians):
- Total People = Total Teachers + Total Librarians = 9 + 36 = 45
#### Step 2: Probability calculations
Next, calculate the individual probabilities required.
1. [tex]\( P(\text{Teacher and chooses Book}) \)[/tex]
- Probability of a person being a teacher who prefers books = [tex]\(\frac{\text{Number of teachers who prefer books}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer books}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Teacher and chooses Book}) = \frac{4}{45} \times \frac{13}{45} = 0.02567901234567901 \)[/tex]
2. [tex]\( P(\text{Librarian and chooses Movie}) \)[/tex]
- Probability of a person being a librarian who prefers movies = [tex]\(\frac{\text{Number of librarians who prefer movies}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer movies}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Librarian and chooses movie}) = \frac{12}{45} \times \frac{15}{45} = 0.08888888888888888 \)[/tex]
3. [tex]\( P(\text{Teacher and chooses Play}) \)[/tex]
- Probability of a person being a teacher who prefers plays = [tex]\(\frac{\text{Number of teachers who prefer plays}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer plays}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Teacher and chooses Play}) = \frac{2}{45} \times \frac{17}{45} = 0.016790123456790124 \)[/tex]
4. [tex]\( P(\text{Librarian and chooses Books}) \)[/tex]
- Probability of a person being a librarian who prefers books = [tex]\(\frac{\text{Number of librarians who prefer books}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer books}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Librarian and chooses Books}) = \frac{9}{45} \times \frac{13}{45} = 0.057777777777777775 \)[/tex]
### Conclusion:
Based on the calculated probabilities:
1. [tex]\( P(\text{Teacher and chooses Book}) = 0.02567901234567901 \)[/tex]
2. [tex]\( P(\text{Librarian and chooses Movie}) = 0.08888888888888888 \)[/tex]
3. [tex]\( P(\text{Teacher and chooses Play}) = 0.016790123456790124 \)[/tex]
4. [tex]\( P(\text{Librarian and chooses Books}) = 0.057777777777777775 \)[/tex]
The analysis shows that all these probabilities fit the pattern of independent events, as their calculations involve the multiplication of the individual probabilities of related but independent events.
Hence, all the given options are examples of independent events.
### Step-by-Step Calculation:
#### Step 1: Total counts of preferences
First, calculate the total number of teachers and librarians:
- Teachers:
- Movies: 3
- Books: 4
- Plays: 2
- Total Teachers = 3 + 4 + 2 = 9
- Librarians:
- Movies: 12
- Books: 9
- Plays: 15
- Total Librarians = 12 + 9 + 15 = 36
Next, calculate the overall totals for each activity:
- Movies:
- Total Movies = 3 (teachers) + 12 (librarians) = 15
- Books:
- Total Books = 4 (teachers) + 9 (librarians) = 13
- Plays:
- Total Plays = 2 (teachers) + 15 (librarians) = 17
- Total People (Teachers + Librarians):
- Total People = Total Teachers + Total Librarians = 9 + 36 = 45
#### Step 2: Probability calculations
Next, calculate the individual probabilities required.
1. [tex]\( P(\text{Teacher and chooses Book}) \)[/tex]
- Probability of a person being a teacher who prefers books = [tex]\(\frac{\text{Number of teachers who prefer books}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer books}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Teacher and chooses Book}) = \frac{4}{45} \times \frac{13}{45} = 0.02567901234567901 \)[/tex]
2. [tex]\( P(\text{Librarian and chooses Movie}) \)[/tex]
- Probability of a person being a librarian who prefers movies = [tex]\(\frac{\text{Number of librarians who prefer movies}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer movies}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Librarian and chooses movie}) = \frac{12}{45} \times \frac{15}{45} = 0.08888888888888888 \)[/tex]
3. [tex]\( P(\text{Teacher and chooses Play}) \)[/tex]
- Probability of a person being a teacher who prefers plays = [tex]\(\frac{\text{Number of teachers who prefer plays}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer plays}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Teacher and chooses Play}) = \frac{2}{45} \times \frac{17}{45} = 0.016790123456790124 \)[/tex]
4. [tex]\( P(\text{Librarian and chooses Books}) \)[/tex]
- Probability of a person being a librarian who prefers books = [tex]\(\frac{\text{Number of librarians who prefer books}}{\text{Total number of people}}\)[/tex] [tex]\(\times\ \frac{\text{Total number of people who prefer books}}{\text{Total number of people}}\)[/tex]
- [tex]\( P(\text{Librarian and chooses Books}) = \frac{9}{45} \times \frac{13}{45} = 0.057777777777777775 \)[/tex]
### Conclusion:
Based on the calculated probabilities:
1. [tex]\( P(\text{Teacher and chooses Book}) = 0.02567901234567901 \)[/tex]
2. [tex]\( P(\text{Librarian and chooses Movie}) = 0.08888888888888888 \)[/tex]
3. [tex]\( P(\text{Teacher and chooses Play}) = 0.016790123456790124 \)[/tex]
4. [tex]\( P(\text{Librarian and chooses Books}) = 0.057777777777777775 \)[/tex]
The analysis shows that all these probabilities fit the pattern of independent events, as their calculations involve the multiplication of the individual probabilities of related but independent events.
Hence, all the given options are examples of independent events.
