Answer :
Let's analyze each statement one by one in detail.
### Statement 1:
The probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
To find the probability of randomly selecting a female student whose favorite genre is drama, we need to use the formula:
[tex]\[ P(F \text{ and } D) = \frac{\text{Number of female students who like drama}}{\text{Total number of students}} \][/tex]
From the table, the number of female students who like drama is 24 and the total number of students is 240. Thus,
[tex]\[ P(F \text{ and } D) = \frac{24}{240} = 0.10 \][/tex]
So, the probability is 0.10 or 10%, not 17%. Therefore, this statement is incorrect.
### Statement 2:
Event [tex]\(F\)[/tex] for female and event [tex]\(D\)[/tex] for drama are independent events.
To check if two events are independent, we need to verify if:
[tex]\[ P(F \text{ and } D) = P(F) \times P(D) \][/tex]
We already calculated [tex]\( P(F \text{ and } D) = \frac{24}{240} = 0.10 \)[/tex].
Next, we'll calculate [tex]\( P(F) \)[/tex] and [tex]\( P(D) \)[/tex]:
[tex]\[ P(F) = \frac{\text{Total number of female students}}{\text{Total number of students}} = \frac{144}{240} = 0.60 \][/tex]
[tex]\[ P(D) = \frac{\text{Total number of students who like drama}}{\text{Total number of students}} = \frac{40}{240} \approx 0.167 \][/tex]
Now, checking for independence:
[tex]\[ P(F) \times P(D) = 0.60 \times 0.167 \approx 0.1002 \][/tex]
Since [tex]\( P(F \text{ and } D) = 0.10 \neq P(F) \times P(D) \approx 0.1002 \)[/tex], the events are not independent. Thus, this statement is incorrect.
### Statement 3:
The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
Conditional probability is given by:
[tex]\[ P(M \mid H) = \frac{P(M \text{ and } H)}{P(H)} \][/tex]
From the table, the number of male students who like horror is 16, and the total number of students who like horror is 38. Thus,
[tex]\[ P(M \mid H) = \frac{16}{38} \approx 0.421 \][/tex]
This result aligns with the calculation, and the given fraction [tex]\(\frac{16}{40}\)[/tex] does not match with the actual data.
Therefore, this statement is incorrect.
### Statement 4:
Event [tex]\(M\)[/tex] for male and event [tex]\(A\)[/tex] for action are independent events.
To check if two events are independent, we use:
[tex]\[ P(M \text{ and } A) = P(M) \times P(A) \][/tex]
Number of males who like action: 28.
Total number of students: 240.
So,
[tex]\[ P(M \text{ and } A) = \frac{28}{240} = 0.117 \][/tex]
Now, calculate [tex]\( P(M) \)[/tex] and [tex]\( P(A) \)[/tex]:
[tex]\[ P(M) = \frac{\text{Total number of male students}}{\text{Total number of students}} = \frac{96}{240} = 0.40 \][/tex]
[tex]\[ P(A) = \frac{\text{Total number of students who like action}}{\text{Total number of students}} = \frac{72}{240} = 0.30 \][/tex]
Checking for independence:
[tex]\[ P(M) \times P(A) = 0.40 \times 0.30 = 0.12 \][/tex]
Since [tex]\( P(M \text{ and } A) = 0.117 \neq P(M) \times P(A) = 0.12 \)[/tex], the events are not independent. Hence, this statement is incorrect.
So, based on these explanations and numerical results:
- Statement 1 is incorrect.
- Statement 2 is incorrect.
- Statement 3 is incorrect.
- Statement 4 is incorrect.
### Statement 1:
The probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
To find the probability of randomly selecting a female student whose favorite genre is drama, we need to use the formula:
[tex]\[ P(F \text{ and } D) = \frac{\text{Number of female students who like drama}}{\text{Total number of students}} \][/tex]
From the table, the number of female students who like drama is 24 and the total number of students is 240. Thus,
[tex]\[ P(F \text{ and } D) = \frac{24}{240} = 0.10 \][/tex]
So, the probability is 0.10 or 10%, not 17%. Therefore, this statement is incorrect.
### Statement 2:
Event [tex]\(F\)[/tex] for female and event [tex]\(D\)[/tex] for drama are independent events.
To check if two events are independent, we need to verify if:
[tex]\[ P(F \text{ and } D) = P(F) \times P(D) \][/tex]
We already calculated [tex]\( P(F \text{ and } D) = \frac{24}{240} = 0.10 \)[/tex].
Next, we'll calculate [tex]\( P(F) \)[/tex] and [tex]\( P(D) \)[/tex]:
[tex]\[ P(F) = \frac{\text{Total number of female students}}{\text{Total number of students}} = \frac{144}{240} = 0.60 \][/tex]
[tex]\[ P(D) = \frac{\text{Total number of students who like drama}}{\text{Total number of students}} = \frac{40}{240} \approx 0.167 \][/tex]
Now, checking for independence:
[tex]\[ P(F) \times P(D) = 0.60 \times 0.167 \approx 0.1002 \][/tex]
Since [tex]\( P(F \text{ and } D) = 0.10 \neq P(F) \times P(D) \approx 0.1002 \)[/tex], the events are not independent. Thus, this statement is incorrect.
### Statement 3:
The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
Conditional probability is given by:
[tex]\[ P(M \mid H) = \frac{P(M \text{ and } H)}{P(H)} \][/tex]
From the table, the number of male students who like horror is 16, and the total number of students who like horror is 38. Thus,
[tex]\[ P(M \mid H) = \frac{16}{38} \approx 0.421 \][/tex]
This result aligns with the calculation, and the given fraction [tex]\(\frac{16}{40}\)[/tex] does not match with the actual data.
Therefore, this statement is incorrect.
### Statement 4:
Event [tex]\(M\)[/tex] for male and event [tex]\(A\)[/tex] for action are independent events.
To check if two events are independent, we use:
[tex]\[ P(M \text{ and } A) = P(M) \times P(A) \][/tex]
Number of males who like action: 28.
Total number of students: 240.
So,
[tex]\[ P(M \text{ and } A) = \frac{28}{240} = 0.117 \][/tex]
Now, calculate [tex]\( P(M) \)[/tex] and [tex]\( P(A) \)[/tex]:
[tex]\[ P(M) = \frac{\text{Total number of male students}}{\text{Total number of students}} = \frac{96}{240} = 0.40 \][/tex]
[tex]\[ P(A) = \frac{\text{Total number of students who like action}}{\text{Total number of students}} = \frac{72}{240} = 0.30 \][/tex]
Checking for independence:
[tex]\[ P(M) \times P(A) = 0.40 \times 0.30 = 0.12 \][/tex]
Since [tex]\( P(M \text{ and } A) = 0.117 \neq P(M) \times P(A) = 0.12 \)[/tex], the events are not independent. Hence, this statement is incorrect.
So, based on these explanations and numerical results:
- Statement 1 is incorrect.
- Statement 2 is incorrect.
- Statement 3 is incorrect.
- Statement 4 is incorrect.
