To find [tex]\( P(C \mid Y) \)[/tex], you need to determine the probability of event [tex]\( C \)[/tex] occurring given that event [tex]\( Y \)[/tex] has occurred. This is calculated using conditional probability, which is defined as:
[tex]\[ P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)} \][/tex]
where:
- [tex]\( P(C \cap Y) \)[/tex] is the joint probability of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] occurring.
- [tex]\( P(Y) \)[/tex] is the probability of [tex]\( Y \)[/tex] occurring.
From the given table, let's identify these values step-by-step:
1. Identify the joint occurrence of [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] (i.e., the value corresponding to [tex]\( C \)[/tex] in row [tex]\( C \)[/tex] and column [tex]\( Y \)[/tex]):
- [tex]\( P(C \cap Y) \)[/tex] is represented by the value 15.
2. Identify the total occurrences of [tex]\( Y \)[/tex] (i.e., the total value in column [tex]\( Y \)[/tex]):
- [tex]\( P(Y) \)[/tex] is represented by the total value 30.
Using these values, calculate [tex]\( P(C \mid Y) \)[/tex]:
[tex]\[ P(C \mid Y) = \frac{P(C \cap Y)}{P(Y)} = \frac{15}{30} = 0.5 \][/tex]
Thus, the value of [tex]\( P(C \mid Y) \)[/tex] to the nearest tenth is:
[tex]\[ \boxed{0.5} \][/tex]
