2
\begin{tabular}{|c|c|c|c|c|}
\hline & [tex]$X$[/tex] & [tex]$Y$[/tex] & [tex]$Z$[/tex] & Total \\
\hline A & 45 & 30 & 60 & 135 \\
\hline B & 20 & 10 & 25 & 55 \\
\hline C & 25 & 35 & 50 & 110 \\
\hline Total & 90 & 75 & 135 & 300 \\
\hline
\end{tabular}

Which statement is true about whether C and Y are independent events?

A. [tex]$C$[/tex] and [tex]$Y$[/tex] are independent events because [tex]$P(C \mid Y) = P(Y)$[/tex].
B. [tex]$C$[/tex] and [tex]$Y$[/tex] are independent events because [tex]$P(C \mid Y) = P(C)$[/tex].
C. [tex]$C$[/tex] and [tex]$Y$[/tex] are not independent events because [tex]$P(C \mid Y) \neq P(Y)$[/tex].
D. [tex]$C$[/tex] and [tex]$Y$[/tex] are not independent events because [tex]$P(C \mid Y) \neq P(C)$[/tex].



Answer :

To determine whether the events [tex]\(C\)[/tex] and [tex]\(Y\)[/tex] are independent, we need to compare the probability [tex]\(P(C \mid Y)\)[/tex] with the probability [tex]\(P(C)\)[/tex].

Let's carefully calculate the required probabilities:

1. Calculate [tex]\(P(C)\)[/tex]:
[tex]\[ P(C) = \frac{\text{Number of outcomes where } C \text{ occurs}}{\text{Total number of outcomes}} = \frac{110}{300} \][/tex]

2. Calculate [tex]\(P(Y)\)[/tex]:
[tex]\[ P(Y) = \frac{\text{Number of outcomes where } Y \text{ occurs}}{\text{Total number of outcomes}} = \frac{75}{300} \][/tex]

3. Calculate [tex]\(P(C \mid Y)\)[/tex]:
[tex]\[ P(C \mid Y) = \frac{\text{Number of outcomes where both } C \text{ and } Y \text{ occur}}{\text{Number of outcomes where } Y \text{ occurs}} = \frac{35}{75} \][/tex]

Once we have these probabilities, we compare [tex]\(P(C \mid Y)\)[/tex] with [tex]\(P(C)\)[/tex]:

[tex]\[ P(C \mid Y) = \frac{35}{75} \][/tex]

[tex]\[ P(C) = \frac{110}{300} = \frac{11}{30} \][/tex]

Next, we simplify [tex]\(\frac{35}{75}\)[/tex]:

[tex]\[ P(C \mid Y) = \frac{35}{75} = \frac{7}{15} \][/tex]

Now let's compare [tex]\(\frac{7}{15}\)[/tex] with [tex]\(\frac{11}{30}\)[/tex].

[tex]\[ \frac{11}{30} \approx 0.3667 \][/tex]
[tex]\[ \frac{7}{15} \approx 0.4667 \][/tex]

Clearly, [tex]\(\frac{7}{15} \neq \frac{11}{30}\)[/tex], hence:

[tex]\(P(C \mid Y) \neq P(C)\)[/tex].

Therefore, the correct statement is:

[tex]\(C\)[/tex] and [tex]\(Y\)[/tex] are not independent events because [tex]\(P(C \mid Y) \neq P(C)\)[/tex].