For a certain population of employees, the percentage

Step 1 of 2:

(a)

In this question, we are asked to find whether the events A and M are independent or not.

The percentage passing or failing a job competency exam, listed according to sex, were as shown in the table.

24% were in the male-pass category, 16% were in the male-fail category.

Let A be the event that the employee pass the exam.

Let M be the event that a male is selected.

\(\begin{array}{|l|l|l|l|}
\hline \text { Outcome } & \text { Male }(M) & \text { Female }(F) & \text { Total } \\
\hline \text { Pass }(A) & 24 & 36 & 60 \\
\hline \text { Fail }(\bar{A}) & 16 & 24 & 40 \\
\hline \text { Total } & 40 & 60 & 100 \\
\hline
\end{array}\)

Probability that the employee scores a passing grade on the exam,

\(P(A)=\frac{60}{100}=0.6[\text { from the table }]\)

Probability that the employee fails on the exam,

\(P(\bar{A})=\frac{40}{100}=0.4[\text { from the table }]\)

Hence we need to find  the conditional probability that the employee scores a passing grade on the exam, given that the employee is male,

\(P(A \mid M)=\frac{P(A \cap M)}{P(M)} \ldots \ldots \ldots(1)\)

From the table,

\(\begin{array}{l}
P(A \cap M)=\frac{24}{100}=0.24\\
P(M)=\frac{40}{100}=0.4\\
P(A \mid M)=\frac{0.24}{0.4}=0.6 .
\end{array} \dots \dots (2)\)

If the events A and M are independent, then we can write,

\(P(A \cap M)=P(A) \times P(M) \ldots \ldots \ldots(3)\)

Therefore we can rewrite the equation (1) ,

\(\begin{array}{l}
P(A \mid M)=\frac{P(A) \times P(M)}{P(M)}=P(A)\\
P(A \mid M)=P(A)=0.6
\end{array} \dots \dots (4)\)

We can see that the  value of  equation (2) and (4) are equal, which means both events are independent.

Hence the events A and M are independent.