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Sagot :
Let's go through each statement one by one and determine their accuracy based on the provided two-way table of gender and favorite genre.
### 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 selecting a female student whose favorite genre is drama, we use the following formula:
[tex]\[ P(\text{Female and Drama}) = \frac{\text{Number of Females who like Drama}}{\text{Total Population}} \][/tex]
From the table, the number of females who like drama is 24 and the total population is 240. Therefore:
[tex]\[ P(\text{Female and Drama}) = \frac{24}{240} = 0.1 \][/tex]
To convert this probability to a percentage:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
This means the probability is 10 percent, not 17 percent. So, this statement is false.
### Statement 2:
Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
To determine if the events are independent, we need to check if:
[tex]\[ P(F \text{ and } D) = P(F) \times P(D) \][/tex]
Where:
- [tex]\( P(F) \)[/tex] is the probability of selecting a female,
- [tex]\( P(D) \)[/tex] is the probability of selecting someone whose favorite genre is drama,
- [tex]\( P(F \text{ and } D) \)[/tex] is the joint probability of both events occurring together.
From the table:
[tex]\[ P(F) = \frac{\text{Number of Females}}{\text{Total Population}} = \frac{144}{240} = 0.6 \][/tex]
[tex]\[ P(D) = \frac{\text{Number of People who like Drama}}{\text{Total Population}} = \frac{40}{240} = 0.1667 \][/tex]
[tex]\[ P(F \text{ and } D) = \frac{24}{240} = 0.1 \][/tex]
Now, let's calculate [tex]\( P(F) \times P(D) \)[/tex]:
[tex]\[ 0.6 \times 0.1667 = 0.10002 \approx 0.1 \][/tex]
So, [tex]\( P(F \text{ and } D) \neq P(F) \times P(D) \)[/tex], which implies the events are not independent. Therefore, this statement is false.
### Statement 3:
The probability of randomly selecting a male student whose favorite genre is horror is [tex]\(\frac{15}{240}\)[/tex].
To find this probability, we use the formula:
[tex]\[ P(\text{Male and Horror}) = \frac{\text{Number of Males who like Horror}}{\text{Total Population}} \][/tex]
From the table:
[tex]\[ P(\text{Male and Horror}) = \frac{16}{240} = 0.0667 \][/tex]
Given probability in the statement is:
[tex]\[ \frac{15}{240} = 0.0625 \][/tex]
Since [tex]\( 0.0667 \neq 0.0625 \)[/tex], this statement is false.
### Statement 4:
Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
To determine if the events are independent, we need to check if:
[tex]\[ P(M \text{ and } A) = P(M) \times P(A) \][/tex]
Where:
- [tex]\( P(M) \)[/tex] is the probability of selecting a male,
- [tex]\( P(A) \)[/tex] is the probability of selecting someone whose favorite genre is action,
- [tex]\( P(M \text{ and } A) \)[/tex] is the joint probability of both events occurring together.
From the table:
[tex]\[ P(M) = \frac{\text{Number of Males}}{\text{Total Population}} = \frac{96}{240} = 0.4 \][/tex]
[tex]\[ P(A) = \frac{\text{Number of People who like Action}}{\text{Total Population}} = \frac{72}{240} = 0.3 \][/tex]
[tex]\[ P(M \text{ and } A) = \frac{28}{240} = 0.1167 \][/tex]
Now, let's calculate [tex]\( P(M) \times P(A) \)[/tex]:
[tex]\[ 0.4 \times 0.3 = 0.12 \][/tex]
So, [tex]\( P(M \text{ and } A) \neq P(M) \times P(A) \)[/tex], which implies the events are not independent. Therefore, this statement is false.
### Conclusion:
All the statements provided are false based on the given data.
### 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 selecting a female student whose favorite genre is drama, we use the following formula:
[tex]\[ P(\text{Female and Drama}) = \frac{\text{Number of Females who like Drama}}{\text{Total Population}} \][/tex]
From the table, the number of females who like drama is 24 and the total population is 240. Therefore:
[tex]\[ P(\text{Female and Drama}) = \frac{24}{240} = 0.1 \][/tex]
To convert this probability to a percentage:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
This means the probability is 10 percent, not 17 percent. So, this statement is false.
### Statement 2:
Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
To determine if the events are independent, we need to check if:
[tex]\[ P(F \text{ and } D) = P(F) \times P(D) \][/tex]
Where:
- [tex]\( P(F) \)[/tex] is the probability of selecting a female,
- [tex]\( P(D) \)[/tex] is the probability of selecting someone whose favorite genre is drama,
- [tex]\( P(F \text{ and } D) \)[/tex] is the joint probability of both events occurring together.
From the table:
[tex]\[ P(F) = \frac{\text{Number of Females}}{\text{Total Population}} = \frac{144}{240} = 0.6 \][/tex]
[tex]\[ P(D) = \frac{\text{Number of People who like Drama}}{\text{Total Population}} = \frac{40}{240} = 0.1667 \][/tex]
[tex]\[ P(F \text{ and } D) = \frac{24}{240} = 0.1 \][/tex]
Now, let's calculate [tex]\( P(F) \times P(D) \)[/tex]:
[tex]\[ 0.6 \times 0.1667 = 0.10002 \approx 0.1 \][/tex]
So, [tex]\( P(F \text{ and } D) \neq P(F) \times P(D) \)[/tex], which implies the events are not independent. Therefore, this statement is false.
### Statement 3:
The probability of randomly selecting a male student whose favorite genre is horror is [tex]\(\frac{15}{240}\)[/tex].
To find this probability, we use the formula:
[tex]\[ P(\text{Male and Horror}) = \frac{\text{Number of Males who like Horror}}{\text{Total Population}} \][/tex]
From the table:
[tex]\[ P(\text{Male and Horror}) = \frac{16}{240} = 0.0667 \][/tex]
Given probability in the statement is:
[tex]\[ \frac{15}{240} = 0.0625 \][/tex]
Since [tex]\( 0.0667 \neq 0.0625 \)[/tex], this statement is false.
### Statement 4:
Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
To determine if the events are independent, we need to check if:
[tex]\[ P(M \text{ and } A) = P(M) \times P(A) \][/tex]
Where:
- [tex]\( P(M) \)[/tex] is the probability of selecting a male,
- [tex]\( P(A) \)[/tex] is the probability of selecting someone whose favorite genre is action,
- [tex]\( P(M \text{ and } A) \)[/tex] is the joint probability of both events occurring together.
From the table:
[tex]\[ P(M) = \frac{\text{Number of Males}}{\text{Total Population}} = \frac{96}{240} = 0.4 \][/tex]
[tex]\[ P(A) = \frac{\text{Number of People who like Action}}{\text{Total Population}} = \frac{72}{240} = 0.3 \][/tex]
[tex]\[ P(M \text{ and } A) = \frac{28}{240} = 0.1167 \][/tex]
Now, let's calculate [tex]\( P(M) \times P(A) \)[/tex]:
[tex]\[ 0.4 \times 0.3 = 0.12 \][/tex]
So, [tex]\( P(M \text{ and } A) \neq P(M) \times P(A) \)[/tex], which implies the events are not independent. Therefore, this statement is false.
### Conclusion:
All the statements provided are false based on the given data.
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