To find [tex]\( P(C \mid Y) \)[/tex], which is the probability of event [tex]\( C \)[/tex] given event [tex]\( Y \)[/tex], we need to follow these steps:
1. Identify the total number of occurrences of event [tex]\( Y \)[/tex].
2. Identify the number of occurrences where both events [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] happen.
3. Use the conditional probability formula:
[tex]\[
P(C \mid Y) = \frac{\text{Number of occurrences of both } C \text{ and } Y}{\text{Total number of occurrences of } Y}
\][/tex]
Looking at the given table, we extract the necessary information:
- The total number of occurrences of [tex]\( Y \)[/tex] is found in the "Total" column under [tex]\( Y \)[/tex]. This is:
[tex]\[
\text{Total}_{Y} = 30
\][/tex]
- The number of occurrences of both [tex]\( C \)[/tex] and [tex]\( Y \)[/tex] is found at the intersection of row [tex]\( C \)[/tex] and column [tex]\( Y \)[/tex]. This is:
[tex]\[
\text{Total}_{C \cap Y} = 15
\][/tex]
Using these values in the conditional probability formula, we calculate:
[tex]\[
P(C \mid Y) = \frac{15}{30} = \frac{1}{2} = 0.5
\][/tex]
To the nearest tenth, the value of [tex]\( P(C \mid Y) \)[/tex] is:
[tex]\[
0.5
\][/tex]
So, the closest value to the answer is:
[tex]\[
0.5
\][/tex]
