Answer :
Let's analyze each statement one by one using the given probabilities and statistics.
1. The probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
- The table shows that there are 24 female students whose favorite genre is drama.
- The total number of students is 240.
- The probability is then calculated as:
[tex]\[ \text{Probability} = \frac{24}{240} = 0.1 = 10 \% \][/tex]
Therefore, this statement is false because the correct probability is 10%, not 17%.
2. Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
For events to be independent, the probability of both events occurring together should be equal to the product of their individual probabilities. Here’s the calculation:
- Probability of being female [tex]\( P(F) \)[/tex]:
[tex]\[ P(F) = \frac{144}{240} = 0.6 \][/tex]
- Probability of liking drama [tex]\( P(D) \)[/tex]:
[tex]\[ P(D) = \frac{40}{240} = 0.167 \][/tex]
- Probability of being female and liking drama [tex]\( P(F \cap D) \)[/tex]:
[tex]\[ P(F \cap D) = \frac{24}{240} = 0.1 \][/tex]
To check for independence:
[tex]\[ P(F) \times P(D) = 0.6 \times 0.167 \approx 0.1002 \][/tex]
Since [tex]\( P(F \cap D) = 0.1 \neq 0.1002 \)[/tex], the events are not exactly independent. Given the rounding, they are very close, but strictly speaking, they are not independent. So, this statement is false.
3. The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
Let's calculate the probability of selecting a male student given that the student's favorite genre is horror:
- Total number of horror fans [tex]\( = 38 \)[/tex]
- Number of male horror fans [tex]\( = 16 \)[/tex]
[tex]\[ P(M \mid H) = \frac{16}{38} \approx 0.421 \][/tex]
This statement says [tex]\(\frac{16}{40}\)[/tex], which is approximately 0.4, not matching the actual calculated value of 0.421. Therefore, this statement is false.
4. Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
For independence, we check if:
[tex]\[ P(M \cap A) = P(M) \times P(A) \][/tex]
Calculations:
- Probability of selecting a male student [tex]\( P(M) \)[/tex]:
[tex]\[ P(M) = \frac{96}{240} = 0.4 \][/tex]
- Probability of selecting a student whose favorite genre is action [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{72}{240} = 0.3 \][/tex]
- Joint probability of a student being male and having action as their favorite genre [tex]\( P(M \cap A) \)[/tex]:
[tex]\[ P(M \cap A) = \frac{28}{240} \approx 0.117 \][/tex]
To check for independence:
[tex]\[ P(M) \times P(A) = 0.4 \times 0.3 = 0.12 \][/tex]
Given [tex]\( P(M \cap A) = 0.117 \)[/tex] and [tex]\( 0.12 \approx 0.117 \)[/tex], the probabilities are very close, though not exact. Based on the provided probabilities:
[tex]\[ P(M \cap A) \approx 0.1167 \][/tex]
This suggests some minor rounding variation, yet the statement strictly implies a substantial level of independence; hence, we can consider the posed events as not independent purely. So, this statement is false.
Conclusion: All given statements are false.
1. The probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
- The table shows that there are 24 female students whose favorite genre is drama.
- The total number of students is 240.
- The probability is then calculated as:
[tex]\[ \text{Probability} = \frac{24}{240} = 0.1 = 10 \% \][/tex]
Therefore, this statement is false because the correct probability is 10%, not 17%.
2. Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
For events to be independent, the probability of both events occurring together should be equal to the product of their individual probabilities. Here’s the calculation:
- Probability of being female [tex]\( P(F) \)[/tex]:
[tex]\[ P(F) = \frac{144}{240} = 0.6 \][/tex]
- Probability of liking drama [tex]\( P(D) \)[/tex]:
[tex]\[ P(D) = \frac{40}{240} = 0.167 \][/tex]
- Probability of being female and liking drama [tex]\( P(F \cap D) \)[/tex]:
[tex]\[ P(F \cap D) = \frac{24}{240} = 0.1 \][/tex]
To check for independence:
[tex]\[ P(F) \times P(D) = 0.6 \times 0.167 \approx 0.1002 \][/tex]
Since [tex]\( P(F \cap D) = 0.1 \neq 0.1002 \)[/tex], the events are not exactly independent. Given the rounding, they are very close, but strictly speaking, they are not independent. So, this statement is false.
3. The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
Let's calculate the probability of selecting a male student given that the student's favorite genre is horror:
- Total number of horror fans [tex]\( = 38 \)[/tex]
- Number of male horror fans [tex]\( = 16 \)[/tex]
[tex]\[ P(M \mid H) = \frac{16}{38} \approx 0.421 \][/tex]
This statement says [tex]\(\frac{16}{40}\)[/tex], which is approximately 0.4, not matching the actual calculated value of 0.421. Therefore, this statement is false.
4. Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
For independence, we check if:
[tex]\[ P(M \cap A) = P(M) \times P(A) \][/tex]
Calculations:
- Probability of selecting a male student [tex]\( P(M) \)[/tex]:
[tex]\[ P(M) = \frac{96}{240} = 0.4 \][/tex]
- Probability of selecting a student whose favorite genre is action [tex]\( P(A) \)[/tex]:
[tex]\[ P(A) = \frac{72}{240} = 0.3 \][/tex]
- Joint probability of a student being male and having action as their favorite genre [tex]\( P(M \cap A) \)[/tex]:
[tex]\[ P(M \cap A) = \frac{28}{240} \approx 0.117 \][/tex]
To check for independence:
[tex]\[ P(M) \times P(A) = 0.4 \times 0.3 = 0.12 \][/tex]
Given [tex]\( P(M \cap A) = 0.117 \)[/tex] and [tex]\( 0.12 \approx 0.117 \)[/tex], the probabilities are very close, though not exact. Based on the provided probabilities:
[tex]\[ P(M \cap A) \approx 0.1167 \][/tex]
This suggests some minor rounding variation, yet the statement strictly implies a substantial level of independence; hence, we can consider the posed events as not independent purely. So, this statement is false.
Conclusion: All given statements are false.
