To find [tex]\( P(C \mid Y) \)[/tex], we need to determine the probability of [tex]\( C \)[/tex] given [tex]\( Y \)[/tex]. This probability can be calculated using the formula for conditional probability:
[tex]\[
P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)}
\][/tex]
Here, [tex]\( P(C \cap Y) \)[/tex] is the probability of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occurring together, and [tex]\( P(Y) \)[/tex] is the probability of [tex]\( Y \)[/tex] occurring.
From the given table:
- The number of occurrences of [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] together is 15.
- The total number of occurrences of [tex]\( Y \)[/tex] is 30.
Thus, we have:
[tex]\[
P(C \mid Y) = \frac{\text{Number of occurrences of } C \text{ and } Y}{\text{Total number of occurrences of } Y} = \frac{15}{30} = 0.5
\][/tex]
To the nearest tenth, the value of [tex]\( P(C \mid Y) \)[/tex] is [tex]\( 0.5 \)[/tex].
Therefore, the correct answer is [tex]\( 0.5 \)[/tex].
