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].
