Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze each statement given in the context of the provided two-way table to determine their validity.
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 this probability, we divide the number of female students whose favorite genre is drama by the total number of students.
[tex]\[ P(\text{Female and Drama}) = \frac{\text{Number of Female Drama Students}}{\text{Total Number of Students}} = \frac{24}{240} = 0.1 \][/tex]
Converting this probability to a percentage:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
So, the probability is actually about 10 percent, not 17 percent. Therefore, this statement is false.
2. Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
To check for independence, we need to compare [tex]\( P(\text{Female} \cap \text{Drama}) \)[/tex] with [tex]\( P(\text{Female}) \times P(\text{Drama}) \)[/tex].
We already have [tex]\( P(\text{Female and Drama}) = 0.1 \)[/tex].
Next, we find [tex]\( P(\text{Female}) \)[/tex] and [tex]\( P(\text{Drama}) \)[/tex]:
[tex]\[ P(\text{Female}) = \frac{\text{Number of Female Students}}{\text{Total Number of Students}} = \frac{144}{240} = 0.6 \][/tex]
[tex]\[ P(\text{Drama}) = \frac{\text{Number of Drama Students}}{\text{Total Number of Students}} = \frac{40}{240} = 0.1667 \][/tex]
Now, calculate [tex]\( P(\text{Female}) \times P(\text{Drama}) \)[/tex]:
[tex]\[ P(\text{Female}) \times P(\text{Drama}) = 0.6 \times 0.1667 = 0.1 \][/tex]
Since [tex]\( P(\text{Female} \cap \text{Drama}) = P(\text{Female}) \times P(\text{Drama}) \)[/tex], the two events are independent. However, the statement is false as it claims they are dependent events.
3. The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
The probability of selecting a male student given that his favorite genre is horror can be found using conditional probability:
[tex]\[ P(\text{Male} | \text{Horror}) = \frac{\text{Number of Male Horror Students}}{\text{Total Number of Horror Students}} = \frac{16}{38} \][/tex]
Simplifying this fraction:
[tex]\[ \frac{16}{38} \approx 0.421 \][/tex]
The given statement claims the probability is [tex]\(\frac{16}{40}\)[/tex], which is approximately 0.4. Since [tex]\(\frac{16}{38}\)[/tex] is approximately 0.421, this is true, making the probability [tex]\(\approx 0.42\)[/tex].
4. Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
To check if the events [tex]\( M \)[/tex] (male) and [tex]\( A \)[/tex] (action) are independent, we need to compare [tex]\( P(\text{Male} \cap \text{Action}) \)[/tex] with [tex]\( P(\text{Male}) \times P(\text{Action}) \)[/tex].
We already have [tex]\( P(\text{Male}) = \frac{96}{240} = 0.4 \)[/tex] and [tex]\( P(\text{Action}) = \frac{72}{240} = 0.3 \)[/tex].
Now, calculate [tex]\( P(\text{Male and Action}) \)[/tex]:
[tex]\[ P(\text{Male and Action}) = \frac{\text{Number of Male Action Students}}{\text{Total Number of Students}} = \frac{28}{240} = 0.1167 \][/tex]
Calculate [tex]\( P(\text{Male}) \times P(\text{Action}) \)[/tex]:
[tex]\[ P(\text{Male}) \times P(\text{Action}) = 0.4 \times 0.3 = 0.12 \][/tex]
Since [tex]\( P(\text{Male and Action}) \neq P(\text{Male}) \times P(\text{Action}) \)[/tex], the two events are not independent. Therefore, the statement is false.
In summary, the probability of randomly selecting a male student, given that his favorite genre is horror, being [tex]\(\frac{16}{40}\)[/tex] is the only correct statement among the options provided.
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 this probability, we divide the number of female students whose favorite genre is drama by the total number of students.
[tex]\[ P(\text{Female and Drama}) = \frac{\text{Number of Female Drama Students}}{\text{Total Number of Students}} = \frac{24}{240} = 0.1 \][/tex]
Converting this probability to a percentage:
[tex]\[ 0.1 \times 100 = 10\% \][/tex]
So, the probability is actually about 10 percent, not 17 percent. Therefore, this statement is false.
2. Event [tex]\( F \)[/tex] for female and event [tex]\( D \)[/tex] for drama are independent events.
To check for independence, we need to compare [tex]\( P(\text{Female} \cap \text{Drama}) \)[/tex] with [tex]\( P(\text{Female}) \times P(\text{Drama}) \)[/tex].
We already have [tex]\( P(\text{Female and Drama}) = 0.1 \)[/tex].
Next, we find [tex]\( P(\text{Female}) \)[/tex] and [tex]\( P(\text{Drama}) \)[/tex]:
[tex]\[ P(\text{Female}) = \frac{\text{Number of Female Students}}{\text{Total Number of Students}} = \frac{144}{240} = 0.6 \][/tex]
[tex]\[ P(\text{Drama}) = \frac{\text{Number of Drama Students}}{\text{Total Number of Students}} = \frac{40}{240} = 0.1667 \][/tex]
Now, calculate [tex]\( P(\text{Female}) \times P(\text{Drama}) \)[/tex]:
[tex]\[ P(\text{Female}) \times P(\text{Drama}) = 0.6 \times 0.1667 = 0.1 \][/tex]
Since [tex]\( P(\text{Female} \cap \text{Drama}) = P(\text{Female}) \times P(\text{Drama}) \)[/tex], the two events are independent. However, the statement is false as it claims they are dependent events.
3. The probability of randomly selecting a male student, given that his favorite genre is horror, is [tex]\(\frac{16}{40}\)[/tex].
The probability of selecting a male student given that his favorite genre is horror can be found using conditional probability:
[tex]\[ P(\text{Male} | \text{Horror}) = \frac{\text{Number of Male Horror Students}}{\text{Total Number of Horror Students}} = \frac{16}{38} \][/tex]
Simplifying this fraction:
[tex]\[ \frac{16}{38} \approx 0.421 \][/tex]
The given statement claims the probability is [tex]\(\frac{16}{40}\)[/tex], which is approximately 0.4. Since [tex]\(\frac{16}{38}\)[/tex] is approximately 0.421, this is true, making the probability [tex]\(\approx 0.42\)[/tex].
4. Event [tex]\( M \)[/tex] for male and event [tex]\( A \)[/tex] for action are independent events.
To check if the events [tex]\( M \)[/tex] (male) and [tex]\( A \)[/tex] (action) are independent, we need to compare [tex]\( P(\text{Male} \cap \text{Action}) \)[/tex] with [tex]\( P(\text{Male}) \times P(\text{Action}) \)[/tex].
We already have [tex]\( P(\text{Male}) = \frac{96}{240} = 0.4 \)[/tex] and [tex]\( P(\text{Action}) = \frac{72}{240} = 0.3 \)[/tex].
Now, calculate [tex]\( P(\text{Male and Action}) \)[/tex]:
[tex]\[ P(\text{Male and Action}) = \frac{\text{Number of Male Action Students}}{\text{Total Number of Students}} = \frac{28}{240} = 0.1167 \][/tex]
Calculate [tex]\( P(\text{Male}) \times P(\text{Action}) \)[/tex]:
[tex]\[ P(\text{Male}) \times P(\text{Action}) = 0.4 \times 0.3 = 0.12 \][/tex]
Since [tex]\( P(\text{Male and Action}) \neq P(\text{Male}) \times P(\text{Action}) \)[/tex], the two events are not independent. Therefore, the statement is false.
In summary, the probability of randomly selecting a male student, given that his favorite genre is horror, being [tex]\(\frac{16}{40}\)[/tex] is the only correct statement among the options provided.
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
