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

Answer:

The probability of randomly selecting a male student, given that his favorite genre is horror, is $\frac{16}{38}=\frac{8}{19}\neq\frac{16}{40}$.

The probability of randomly selecting a student who has a favorite genre of drama and is also female is $\frac{24}{240}= 0.1 = 10%\neq17%$.

For two events $A$ and $B$ to be independent, $P(A\cap B)=P(A)\times P(B)$.

$P(F)=\frac{144}{240}=\frac{3}{5}$, $P(D)=\frac{40}{240}=\frac{1}{6}$, $P(F\cap D)=\frac{24}{240}=\frac{1}{10}$, and $P(F)\times P(D)=\frac{3}{5}\times\frac{1}{6}=\frac{1}{10}$, so event $F$ for female and event $D$ for drama are independent events.

$P(M)=\frac{96}{240}=\frac{2}{5}$, $P(A)=\frac{72}{240}=\frac{3}{10}$, $P(M\cap A)=\frac{28}{240}=\frac{7}{60}$, and $P(M)\times P(A)=\frac{2}{5}\times\frac{3}{10}=\frac{3}{25}\neq\frac{7}{60}$, so event $M$ for male and event $A$ for action are not independent events.

So the true statement is: Event $F$ for female and event $D$ for drama are independent events.

Explanation:

Step1: Calculate probabilities for drama - female

$P(\text{Female and Drama})=\frac{\text{Number of female - drama}}{\text{Total number of students}}=\frac{24}{240} = 0.1$

Step2: Check independence of $F$ and $D$

$P(F)=\frac{144}{240}$, $P(D)=\frac{40}{240}$, $P(F\cap D)=\frac{24}{240}$, $P(F)\times P(D)=\frac{144}{240}\times\frac{40}{240}=\frac{1}{10}=P(F\cap D)$

Step3: Calculate conditional probability for male - horror

$P(\text{Male}|\text{Horror})=\frac{\text{Number of male - horror}}{\text{Number of horror}}=\frac{16}{38}$

Step4: Check independence of $M$ and $A$

$P(M)=\frac{96}{240}$, $P(A)=\frac{72}{240}$, $P(M\cap A)=\frac{28}{240}$, $P(M)\times P(A)=\frac{96}{240}\times\frac{72}{240}\neq\frac{28}{240}$