Let's analyze the given table and answer the question step-by-step.
First, we need to find the relevant probabilities:
1. [tex]\[ P(C | M^{\prime}) \][/tex]: This is the probability that a person is color-blind given that the person is not male (female).
2. [tex]\[ P(C) \][/tex]: This is the overall probability that a person is color-blind.
We are given the following data in the table:
- [tex]\[ P(C \cap M^{\prime}) = 0.002 \][/tex]
- [tex]\[ P(M^{\prime}) = 0.583 \][/tex]
- [tex]\[ P(C) = 0.034 \][/tex]
First, let's calculate [tex]\[ P(C | M^{\prime}) \][/tex]:
[tex]\[
P(C | M^{\prime}) = \frac{P(C \cap M^{\prime})}{P(M^{\prime})}
\][/tex]
Substituting the given values:
[tex]\[
P(C | M^{\prime}) = \frac{0.002}{0.583} \approx 0.003
\][/tex]
Now we compare [tex]\[ P(C | M^{\prime}) \][/tex] with [tex]\[ P(C) \][/tex]:
[tex]\[
P(C) = 0.034
\][/tex]
Finally, we determine if the events \(C\) and \(M^{\prime}\) are dependent. Two events are dependent if \(P(E \mid F) \neq P(E)\). Here, we compare \(P(C | M^{\prime})\) and \(P(C)\):
[tex]\[
P(C | M^{\prime}) \approx 0.003
\][/tex]
[tex]\[
P(C) = 0.034
\][/tex]
Since \(P(C | M^{\prime}) \neq P(C)\), we conclude that the events \(C\) and \(M^{\prime}\) are dependent.
### Final Answer:
The events [tex]\(C\)[/tex] and [tex]\(M^{\prime}\)[/tex] are dependent because [tex]\(P(C \mid M^{\prime}) = 0.003\)[/tex] and [tex]\(P(C) = 0.034\)[/tex].
