The table shows frequencies for red-green color blindness, where \(M\) represents "person is male" and \(C\) represents "person is color-blind." Are the events \(C\) and \(M^{\prime}\) dependent? Recall that two events \(E\) and \(F\) are dependent if \(P(E \mid F) \neq P(E)\).

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & \(M\) & \(M ^{\prime}\) & Totals \\
\hline \(C\) & 0.032 & 0.002 & 0.034 \\
\(C ^{\prime}\) & 0.385 & 0.581 & 0.966 \\
\hline Totals & 0.417 & 0.583 & 1.000 \\
\hline
\end{tabular}
\][/tex]

The events \(C\) and \(M^{\prime}\) \(\square\) dependent because \(P\left(C \mid M^{\prime}\right) =\) \(\square\) and \(P(C) =\) \(\square\).

(Type integers or decimals rounded to three decimal places as needed.)



Answer :

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].