At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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.
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
