To determine whether the events [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] are independent, we can use the concept of conditional probability. Two events are independent if the probability of one event occurring, given that the other event has occurred, is equal to the probability of the first event occurring alone.
Given data from the table:
- Total population = 300
- Number of occurrences of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] ([tex]\( C \cap Y \)[/tex]) = 35
- Total occurrences of [tex]\( C \)[/tex] = 110
- Total occurrences of [tex]\( Y \)[/tex] = 75
First, let's calculate the probabilities involved:
1. Calculate [tex]\( P(C) \)[/tex]:
[tex]\[ P(C) = \frac{\text{Total occurrences of } C}{\text{Total population}} = \frac{110}{300} \approx 0.3667 \][/tex]
2. Calculate [tex]\( P(Y) \)[/tex]:
[tex]\[ P(Y) = \frac{\text{Total occurrences of } Y}{\text{Total population}} = \frac{75}{300} = 0.25 \][/tex]
3. Calculate [tex]\( P(C \cap Y) \)[/tex]:
[tex]\[ P(C \cap Y) = \frac{\text{Number of occurrences of both } C \text{ and } Y}{\text{Total population}} = \frac{35}{300} \approx 0.1167 \][/tex]
4. Calculate [tex]\( P(C \mid Y) \)[/tex]:
[tex]\[ P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)} = \frac{0.1167}{0.25} \approx 0.4667 \][/tex]
Now, to determine independence, check if [tex]\( P(C \mid Y) \)[/tex] equals [tex]\( P(C) \)[/tex]:
- [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], [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] are not independent events.
Thus, 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].
