Question

a news portal surveyed registered users about whether they prefer to get their news from text articles or from videos on the portal. the table shows the data about respondents ages and preferences.

	age below 20	age 20 or above	total
videos	32	90	122
total	48	135	183

which statement is correct?

a. a respondent preferring videos and a respondent being younger than 20 are dependent events.

b. a respondent preferring text articles and a respondent being 20 or older are dependent events.

c. a respondent preferring videos and a respondent preferring text articles are independent events.

d. a respondent preferring text articles and a respondent being younger than 20 are independent events.

Explanation:

Step1: Define independent event rule

Two events \(A\) and \(B\) are independent if \(P(A \cap B) = P(A) \times P(B)\). If not, they are dependent.

Step2: Test Option A

Let \(A=\)prefers videos, \(B=\)age below 20.
\(P(A)=\frac{122}{183}\), \(P(B)=\frac{48}{183}\), \(P(A \cap B)=\frac{32}{183}\)
Calculate \(P(A) \times P(B) = \frac{122 \times 48}{183^2} = \frac{5856}{33489} \approx 0.175\)
\(P(A \cap B)=\frac{32}{183} \approx 0.175\). So \(P(A \cap B)=P(A) \times P(B)\), events are independent. A is wrong.

Step3: Test Option B

Let \(A=\)prefers text, \(B=\)age 20+.
\(P(A)=\frac{61}{183}\), \(P(B)=\frac{135}{183}\), \(P(A \cap B)=\frac{45}{183}\)
Calculate \(P(A) \times P(B) = \frac{61 \times 135}{183^2} = \frac{8235}{33489} \approx 0.246\)
\(P(A \cap B)=\frac{45}{183} \approx 0.246\). So \(P(A \cap B)=P(A) \times P(B)\), events are independent. B is wrong.

Step4: Test Option C

Let \(A=\)prefers videos, \(B=\)prefers text. These are mutually exclusive events (a user can't prefer both), so \(P(A \cap B)=0\). \(P(A) \times P(B)
eq 0\), so they are dependent. C is wrong.

Step5: Test Option D

Let \(A=\)prefers text, \(B=\)age below 20.
\(P(A)=\frac{61}{183}\), \(P(B)=\frac{48}{183}\), \(P(A \cap B)=\frac{16}{183}\)
Calculate \(P(A) \times P(B) = \frac{61 \times 48}{183^2} = \frac{2928}{33489} \approx 0.087\)
\(P(A \cap B)=\frac{16}{183} \approx 0.087\). So \(P(A \cap B)=P(A) \times P(B)\), events are independent. D is correct.

Answer:

D. A respondent preferring text articles and a respondent being younger than 20 are independent events.