Question

the two - way table shows the distribution of gender to favorite film genre for the senior class at mt. rose high school.

	m	f	total
horror	16	22	38
drama	16	24	40
action	28	44	72
total	96	144	240

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.

Explanation:

To solve this, we analyze each option using the two - way table:

Option 1: Probability of female and drama

• Total number of students, \(N = 240\).
• Number of female students who like drama, \(n(F\cap D)=24\).
• Probability \(P(F\cap D)=\frac{24}{240}=0.1\) or \(10\%\), not \(17\%\). So this option is incorrect.

Option 2: Probability of male and horror (conditional probability)

• Number of male students, \(n(M) = 96\).
• Number of male students who like horror, \(n(M\cap H)=16\).
• The conditional probability \(P(H|M)=\frac{n(M\cap H)}{n(M)}=\frac{16}{96}=\frac{1}{6}

eq\frac{16}{40}\). So this option is incorrect.

Option 3: Independence of female (F) and drama (D)

• \(P(F)=\frac{144}{240} = 0.6\)
• \(P(D)=\frac{40}{240}=\frac{1}{6}\approx0.167\)
• \(P(F\cap D)=\frac{24}{240} = 0.1\)
• For independent events, \(P(F)\times P(D)=0.6\times0.167\approx0.1\), which is equal to \(P(F\cap D)\). So event \(F\) (female) and event \(D\) (drama) are independent events. This option is correct.

Option 4: Independence of male (M) and action (A)

• \(P(M)=\frac{96}{240}=0.4\)
• \(P(A)=\frac{72}{240} = 0.3\)
• \(P(M\cap A)=\frac{28}{240}\approx0.117\)
• \(P(M)\times P(A)=0.4\times0.3 = 0.12

eq0.117\). So event \(M\) (male) and event \(A\) (action) are not independent events. This option is incorrect.

Answer:

The statement "Event F for female and event D for drama are independent events" is true.