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 |
|--|--|--|--|--|
| text articles | 16 | 45 | 61 |
| 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 younger than 20 are independent events.
c. a respondent preferring text articles and a respondent being 20 or older are dependent events.
d. a respondent preferring videos and a respondent preferring text articles are independent events.

Question

Accepted Answer

# Explanation:
## Step1: Recall the definition of independent events
Two events $A$ and $B$ are independent if $P(A\cap B)=P(A)\times P(B)$. Let $A$ be the event of preferring videos and $B$ be the event of being younger than 20.
## Step2: Calculate $P(A)$, $P(B)$ and $P(A\cap B)$
The total number of respondents $n = 183$.
The number of respondents who prefer videos $n(A)=122$, so $P(A)=\frac{122}{183}$.
The number of respondents younger than 20 $n(B) = 48$, so $P(B)=\frac{48}{183}$.
The number of respondents who prefer videos and are younger than 20 $n(A\cap B)=32$, so $P(A\cap B)=\frac{32}{183}$.
And $P(A)\times P(B)=\frac{122}{183}\times\frac{48}{183}=\frac{122\times48}{183\times183}=\frac{5856}{33489}\approx0.175$, while $\frac{32}{183}\approx0.175$. Since $P(A\cap B)
eq P(A)\times P(B)$, they are dependent events.
Let's check other cases in a similar way.
For the event of preferring text - articles ($C$) and being younger than 20 ($B$):
$n(C) = 61$, $P(C)=\frac{61}{183}$, $P(B)=\frac{48}{183}$, $n(C\cap B)=16$, $P(C\cap B)=\frac{16}{183}$, $P(C)\times P(B)=\frac{61}{183}\times\frac{48}{183}=\frac{2928}{33489}
eq\frac{16}{183}$, so they are dependent.
For the event of preferring text - articles ($C$) and being 20 or older ($D$):
$n(C) = 61$, $P(C)=\frac{61}{183}$, $n(D)=135$, $P(D)=\frac{135}{183}$, $n(C\cap D)=45$, $P(C\cap D)=\frac{45}{183}$, $P(C)\times P(D)=\frac{61}{183}\times\frac{135}{183}=\frac{8235}{33489}
eq\frac{45}{183}$, so they are dependent.
For the event of preferring videos ($A$) and preferring text - articles ($C$), $A\cap C=\varnothing$, $P(A\cap C) = 0$, $P(A)\times P(C)=\frac{122}{183}\times\frac{61}{183}
eq0$, so they are dependent.

# Answer:
A. A respondent preferring videos and a respondent being younger than 20 are dependent events.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the definition of independent events

Step2: Calculate \(P(A)\), \(P(B)\) and \(P(A\cap B)\)

Answer: