Answer :
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.
