the two - way table shows the distribution of gender to favorite film genre for the senior class at mt. rose high school.
which statement is true?
the probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
event f for female and event d for drama are independent events.
the probability of randomly selecting a male student, given that his favorite genre is horror, is $\frac{16}{40}$
event m for male and event a for action are independent events.

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Accepted Answer

# Brief Explanations:
1. Analyze the first option: The number of female students who like drama is 24, and the total number of students is 240. The probability is $\frac{24}{240} = 0.1 = 10\%$, not 17%, so this option is wrong.
2. Analyze the second option: For events F (female) and D (drama) to be independent, $P(F\cap D)=P(F)	imes P(D)$. $P(F)=\frac{144}{240}$, $P(D)=\frac{40}{240}$, $P(F)	imes P(D)=\frac{144}{240}	imes\frac{40}{240}=\frac{144	imes40}{240	imes240}=\frac{5760}{57600}=0.1$. $P(F\cap D)=\frac{24}{240}=0.1$. But wait, let's check again. Wait, $P(F)=\frac{144}{240}=0.6$, $P(D)=\frac{40}{240}=\frac{1}{6}\approx0.1667$, $P(F)	imes P(D)=0.6	imes\frac{1}{6}=0.1$, and $P(F\cap D)=\frac{24}{240}=0.1$. But wait, maybe I made a mistake. Wait, no, let's check the third option. The number of male students who like horror is 16, and the total number of students who like horror is 38. So the probability of selecting a male given horror is $\frac{16}{38}$, not $\frac{16}{40}$, so this option is wrong.
3. Analyze the fourth option: For events M (male) and A (action) to be independent, $P(M\cap A)=P(M)	imes P(A)$. $P(M)=\frac{96}{240}=0.4$, $P(A)=\frac{72}{240}=0.3$, $P(M)	imes P(A)=0.4	imes0.3 = 0.12$. $P(M\cap A)=\frac{28}{240}\approx0.1167\approx0.12$? Wait, no, $\frac{28}{240}=\frac{7}{60}\approx0.1167$, and $0.4	imes0.3 = 0.12$. Wait, maybe my calculation is wrong. Wait, no, let's recalculate. $P(M)=\frac{96}{240}=0.4$, $P(A)=\frac{72}{240}=0.3$, so $P(M)	imes P(A)=0.4	imes0.3 = 0.12$. $P(M\cap A)=\frac{28}{240}=\frac{7}{60}\approx0.1167$, which is approximately 0.12? Wait, no, maybe I made a mistake in the second option. Wait, let's re - check the second option. $P(F)=\frac{144}{240}=0.6$, $P(D)=\frac{40}{240}=\frac{1}{6}\approx0.1667$, $P(F\cap D)=\frac{24}{240}=0.1$. $0.6	imes0.1667\approx0.1$, which is equal to $P(F\cap D)$. But wait, the third option: the number of male students who like horror is 16, and the total number of students who like horror is 38 (16 + 22), so the conditional probability $P(M|Horror)=\frac{16}{38}$, not $\frac{16}{40}$, so the third option is wrong. The first option: $\frac{24}{240}=10\%$, wrong. Now, let's check the fourth option again. $P(M)=\frac{96}{240}=0.4$, $P(A)=\frac{72}{240}=0.3$, $P(M\cap A)=\frac{28}{240}=\frac{7}{60}\approx0.1167$. $0.4	imes0.3 = 0.12$, which is approximately equal? Wait, no, maybe I made a mistake. Wait, the correct way: For independence, $P(M|A)=P(M)$. $P(M|A)=\frac{28}{72}=\frac{7}{18}\approx0.3889$, and $P(M)=\frac{96}{240}=0.4$. These are approximately equal? Wait, no, $\frac{7}{18}\approx0.3889$, $0.4 = \frac{2}{5}=0.4$. Close, but not exactly. Wait, maybe the second option was miscalculated. Wait, $P(F)=\frac{144}{240}=0.6$, $P(D)=\frac{40}{240}=\frac{1}{6}\approx0.1667$, $P(F\cap D)=\frac{24}{240}=0.1$. $0.6	imes0.1667\approx0.1$, which is equal. But the third option: the number of male students who like horror is 16, total horror is 38 (16 + 22), so $P(M|Horror)=\frac{16}{38}$, not $\frac{16}{40}$, so third option is wrong. The first option: 24/240 = 10%, wrong. Now, let's check the fourth option again. $P(M)=\frac{96}{240}=0.4$, $P(A)=\frac{72}{240}=0.3$, $P(M\cap A)=\frac{28}{240}=\frac{7}{60}\approx0.1167$. $0.4	imes0.3 = 0.12$, which is approximately equal? Wait, maybe the correct answer is the fourth option? Wait, no, let's recalculate $P(M|A)=\frac{28}{72}=\frac{7}{18}\approx0.3889$, $P(M)=\frac{96}{240}=0.4$. The difference is due to rounding. Wait, maybe the second option is wrong. Wait, no, let's check the definition of independent events: $P(A\cap B)=P(A)P(B)$. For event M (male) and A (action): $P(M)=\frac{96}{240}=0.4$, $P(A)=\frac{72}{240}=0.3$, $P(M\cap A)=\frac{28}{240}=\frac{7}{60}\approx0.1167$. $0.4	imes0.3 = 0.12$, and $\frac{7}{60}\approx0.1167$, which is approximately equal (due to rounding). But the third option: the number of male students who like horror is 16, total horror is 38, so $P(M|Horror)=\frac{16}{38}$, not $\frac{16}{40}$, so third option is wrong. The first option: 24/240 = 10%, wrong. So the correct answer is the fourth option? Wait, no, maybe I made a mistake. Wait, the correct answer is: Let's check each option again.

Option 1: Female and drama: 24/240 = 0.1 = 10% ≠ 17%, wrong.

Option 2: $P(F)=\frac{144}{240}=0.6$, $P(D)=\frac{40}{240}=\frac{1}{6}\approx0.1667$, $P(F\cap D)=\frac{24}{240}=0.1$. $0.6	imes0.1667\approx0.1$, so $P(F\cap D)=P(F)P(D)$, so they are independent? But wait, the number of female and drama is 24, total female 144, total drama 40. 144*40 = 5760, 24*240 = 5760. So by the formula for independence in two - way tables, if (row total * column total)/grand total = cell count, then independent. For F (female row) and D (drama column): 144*40/240 = (144*40)/240 = 144/6 = 24, which is equal to the cell count (24). So F and D are independent? But then why is the fourth option considered? Wait, for M (male row) and A (action column): 96*72/240=(96*72)/240=(96/240)*72 = 0.4*72 = 28.8, but the cell count is 28, which is not equal to 28.8. So M and A are not independent. For the third option: the number of male students who like horror is 16, total horror is 38 (16 + 22), so $P(M|Horror)=\frac{16}{38}$, not $\frac{16}{40}$, so third option is wrong. The first option: 24/240 = 10%, wrong. So the second option: Event F for female and event D for drama are independent events, is correct? Wait, but earlier calculation with the formula (row total * column total)/grand total = cell count: 144*40/240 = 24, which is the cell count for F and D. So they are independent. But wait, the fourth option: 96*72/240 = 28.8≠28, so not independent. The third option: 16/38≠16/40, wrong. First option: 10%≠17%, wrong. So the correct answer is the second option? Wait, no, wait the problem's options: let's re - check the second option: "Event F for female and event D for drama are independent events." And according to the two - way table independence rule, (row total * column total)/grand total = cell count, so 144 (female total) * 40 (drama total)/240 (grand total)= (144*40)/240 = 24, which is the cell count (female and drama). So they are independent. But wait, when I calculated $P(F\cap D)=P(F)P(D)$, it also holds. So the second option is correct? But wait, the initial analysis of the third option: the number of male students who like horror is 16, total horror is 38 (16 + 22), so $P(M|Horror)=\frac{16}{38}$, not $\frac{16}{40}$, so third option is wrong. The first option: 24/240 = 10%, wrong. The fourth option: 96*72/240 = 28.8≠28, so not independent. So the correct answer is the second option: Event F for female and event D for drama are independent events. Wait, but maybe I made a mistake. Wait, the total number of students is 240. Female total: 144, Drama total: 40, Female and Drama: 24. $P(F)=\frac{144}{240}=0.6$, $P(D)=\frac{40}{240}=\frac{1}{6}\approx0.1667$, $P(F\cap D)=\frac{24}{240}=0.1$. $0.6	imes0.1667\approx0.1$, so $P(F\cap D)=P(F)P(D)$, so they are independent. So the correct answer is the second option.
# Answer:
Event F for female and event D for drama are independent events. (The option: Event F for female and event D for drama are independent events.)

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