To determine whether events [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] are independent, we need to compare the conditional probability [tex]\( P(C \mid Y) \)[/tex] with the probability [tex]\( P(C) \)[/tex]. Two events are independent if and only if [tex]\( P(C \mid Y) = P(C) \)[/tex].
Let's break down the calculations step by step:
1. Total number of outcomes:
[tex]\[
\text{Total outcomes} = 300
\][/tex]
2. Total outcomes where [tex]\( C \)[/tex] occurs:
[tex]\[
\text{Total } C = 110
\][/tex]
3. Total outcomes where [tex]\( Y \)[/tex] occurs:
[tex]\[
\text{Total } Y = 75
\][/tex]
4. Outcomes where both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occur:
[tex]\[
\text{C and Y} = 35
\][/tex]
5. Calculate [tex]\( P(C) \)[/tex]:
The probability of [tex]\( C \)[/tex] occurring is given by the ratio of outcomes where [tex]\( C \)[/tex] occurs to the total number of outcomes:
[tex]\[
P(C) = \frac{\text{Total } C}{\text{Total outcomes}} = \frac{110}{300} \approx 0.3667
\][/tex]
6. Calculate [tex]\( P(Y) \)[/tex]:
The probability of [tex]\( Y \)[/tex] occurring is given by the ratio of outcomes where [tex]\( Y \)[/tex] occurs to the total number of outcomes:
[tex]\[
P(Y) = \frac{\text{Total } Y}{\text{Total outcomes}} = \frac{75}{300} = 0.25
\][/tex]
7. Calculate [tex]\( P(C \text{ and } Y) \)[/tex]:
The probability of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occurring is given by the ratio of outcomes where both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occur to the total number of outcomes:
[tex]\[
P(C \text{ and } Y) = \frac{\text{C and Y}}{\text{Total outcomes}} = \frac{35}{300} \approx 0.1167
\][/tex]
8. Calculate [tex]\( P(C \mid Y) \)[/tex]:
The conditional probability [tex]\( P(C \mid Y) \)[/tex] is given by the ratio of the probability of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occurring to the probability of [tex]\( Y \)[/tex] occurring:
[tex]\[
P(C \mid Y) = \frac{P(C \text{ and } Y)}{P(Y)} = \frac{0.1167}{0.25} \approx 0.4667
\][/tex]
9. Check for independence:
To determine if [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] are independent, we compare [tex]\( P(C) \)[/tex] with [tex]\( P(C \mid Y) \)[/tex]. If [tex]\( P(C \mid Y) = P(C) \)[/tex], they are independent. Otherwise, they are not.
From our calculations:
[tex]\[
P(C) \approx 0.3667
\][/tex]
[tex]\[
P(C \mid Y) \approx 0.4667
\][/tex]
Since [tex]\( P(C \mid Y) \neq P(C) \)[/tex], we conclude:
[tex]\[
\text{C and Y are not independent events because } P(C \mid Y) \neq P(C).
\][/tex]
Therefore, the correct statement is:
C and Y are not independent events because [tex]\( P(C \mid Y) \neq P(C) \)[/tex].
