Answered

The two-way table shows the distribution of gender to favorite film genre for the senior class at Mt. Rose High School.




ComedyHorrorDramaActionTotal

M		3616162896

F		54222444144

Total		90384072240






Which statement is true?


AThe probability of randomly selecting a student who has a favorite genre of drama and is also female is about 17 percent.
BEvent F for female and event D for drama are independent events.
CThe probability of randomly selecting a male student, given that his favorite genre is horror, is $\frac{16}{40}$.
DEvent $M$ for male and event $A$ for action are independent events.

Verified Answer :

  • The probability of selecting a student who is female and prefers drama is \(\frac{24}{240} = 10\%\).
  • The conditional probability \(P(M | \text{Horror})\) is \(\frac{16}{38} \approx 42\%\).
  • Events \(M\) and \(A\) are not independent as \(P(M \cap A) \approx 0.1167\) while \(P(M) \cdot P(A) = 0.12\).
  • Events \(F\) and \(D\) are independent because \(P(F \cap D) = P(F) \cdot P(D) = 0.1\).
  • The correct statement is: \(\boxed{\text{Event } F \text{ for female and event } D \text{ for drama are independent events.}}\)

Explanation

  1. Analyze and verify each statement

To determine which statement is true, we first calculate the total probabilities and relevant values from the provided two-way table. The table includes 240 senior students in total. We have 96 males and 144 females. The distribution across genres is: Comedy (90), Horror (38), Drama (40), and Action (72).

  • Probability of a student being female and preferring drama: There are 24 students who are both female and prefer drama. The probability is \(\frac{24}{240} = 0.10 = 10\%\). The statement claiming it is about 17% is false.
  • Probability of a male given their favorite genre is horror: There are 38 students who prefer horror, and 16 of them are male. Thus, \(P(M | \text{Horror}) = \frac{16}{38} \approx 0.42\). The statement claiming it is \(\frac{16}{40}\) is false.
  • Testing the independence of events \(M\) (male) and \(A\) (action): Two events \(X\) and \(Y\) are independent if \(P(X \cap Y) = P(X) \cdot P(Y)\).
  • \(P(M) = \frac{96}{240} = 0.4\)
  • \(P(A) = \frac{72}{240} = 0.3\)
  • \(P(M) \cdot P(A) = 0.4 \cdot 0.3 = 0.12\)
  • \(P(M \cap A) = \frac{28}{240} \approx 0.1167\)
  • Since \(0.12 eq 0.1167\), events \(M\) and \(A\) are not independent.
  • Testing the independence of events \(F\) (female) and \(D\) (drama):
  • \(P(F) = \frac{144}{240} = 0.6\)
  • \(P(D) = \frac{40}{240} = \frac{1}{6} \approx 0.1667\)
  • \(P(F) \cdot P(D) = 0.6 \cdot \frac{1}{6} = 0.1\)
  • \(P(F \cap D) = \frac{24}{240} = 0.1\)
  • Since \(P(F \cap D) = P(F) \cdot P(D) = 0.1\), events \(F\) and \(D\) are independent.

Examples

When analyzing school survey data, knowing if two attributes are independent—like gender and film preference—can help researchers understand if students' tastes are influenced by their gender or are distributed uniformly. For instance, if \(P(F \cap D) = P(F) \cdot P(D)\), we can conclude that for this population, being female does not change the probability of preferring drama, revealing a lack of bias in this specific interest. This statistical tool is vital for unbiased decision-making and interpreting demographic data accurately.

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