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(C) [tex]$\quad P(B / A)=P(B)$[/tex]
(D) [tex]$\quad P\left(A^{\prime} \cap B\right)=P\left(A^{\prime}\right) \cdot P(B)$[/tex]

The table below shows the distribution of students by sex and department in three departments in a university. Use the information to answer questions 27 to 30.

\begin{tabular}{|l|l|l|l|}
\hline
\multirow{2}{*}{Sex} & \multicolumn{3}{c|}{Department} \\
\cline{2-4} & Mathematics & Physics & Statistics \\
\hline
Male & 30 & 68 & 45 \\
\hline
Female & 45 & 22 & 25 \\
\hline
\end{tabular}

27. What is the probability that a student picked at random is from the Mathematics department?
(A) 0.309
(B) 0.329
(C) 0.319
(D) None of these

28. What is the probability that a student picked at random is a male student from the Statistics department?
(A) 0.645
(B) 0.329
(C) 0.649
(D) None of these

29. What is the probability that a student picked at random is not from the Physics department?
(A) 0.617
(B) 0.329
(C) 0.629
(D) None of these

30. What is the probability that a student picked at random is a female?
(A) 0.391
(B) 0.329
(C) 0.392
(D) None of these



Answer :

GinnyAnswer
To solve each of the probability questions, we need to use the information from the table provided:

[tex]\[ \begin{array}{|l|l|l|l|} \hline \multirow{2}{*}{ \text{Sex} } & \text{Department} \\ \cline { 2 - 4 } & \text{Mathematics} & \text{Physics} & \text{Statistics} \\ \hline \text{Male} & 30 & 68 & 45 \\ \hline \text{Female} & 45 & 22 & 25 \\ \hline \end{array} \][/tex]

To find the total number of students, we sum all the entries in the table:

[tex]\[ \text{Total students} = 30 + 45 + 68 + 22 + 45 + 25 = 235 \][/tex]

Now, let's answer each question step-by-step.

### Question 27: Probability that a student picked at random is from the Mathematics department

To find this probability, we need to find the total number of students in the Mathematics department and divide it by the total number of students.

[tex]\[ \text{Total Mathematics students} = 30 \, (\text{Male}) + 45 \, (\text{Female}) = 75 \][/tex]

Therefore, the probability is:

[tex]\[ P(\text{Math}) = \frac{75}{235} \approx 0.319 \][/tex]

The correct option is:
(C) 0.319

### Question 28: Probability that a student picked at random is a male student from the Statistics department

To find this probability, we need to find the number of male students in the Statistics department and divide it by the total number of students.

[tex]\[ \text{Male students in Statistics} = 45 \][/tex]

Therefore, the probability is:

[tex]\[ P(\text{Male Statistics}) = \frac{45}{235} \approx 0.191 \][/tex]

The correct option is:
(D) None of these

### Question 29: Probability that a student picked at random is not from the Physics department

To find this probability, we need to find the total number of students not in the Physics department and divide it by the total number of students.

[tex]\[ \text{Total Physics students} = 68 \, (\text{Male}) + 22 \, (\text{Female}) = 90 \][/tex]

[tex]\[ \text{Students not in Physics} = 235 - 90 = 145 \][/tex]

Therefore, the probability is:

[tex]\[ P(\text{Not Physics}) = \frac{145}{235} \approx 0.617 \][/tex]

The correct option is:
(A) 0.617

### Question 30: Probability that a student picked at random is a female

To find this probability, we need to find the total number of female students and divide it by the total number of students.

[tex]\[ \text{Total female students} = 45 \, (\text{Mathematics}) + 22 \, (\text{Physics}) + 25 \, (\text{Statistics}) = 92 \][/tex]

Therefore, the probability is:

[tex]\[ P(\text{Female}) = \frac{92}{235} \approx 0.391 \][/tex]

The correct option is:
(A) 0.391

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