Questions: A survey was taken to determine the birthplace of a class of students in a statistics class. Below is the data set. Gender Number of Students Birthplace --------- Male 6 San Diego Female 10 San Diego Male 12 Los Angeles Female 8 Los Angeles Male 5 San Francisco Female 9 San Francisco a. What is the probability that a student was born in Los Angeles given the student was male? (3 pts) b. What is the probability a student was born in San Diego or the student was female? (3 pts)

A survey was taken to determine the birthplace of a class of students in a statistics class. Below is the data set. Gender Number of Students Birthplace --------- Male 6 San Diego Female 10 San Diego Male 12 Los Angeles Female 8 Los Angeles Male 5 San Francisco Female 9 San Francisco a. What is the probability that a student was born in Los Angeles given the student was male? (3 pts) b. What is the probability a student was born in San Diego or the student was female? (3 pts)
Transcript text: 3. A survey was taken to determine the birthplace of a class of students in a statistics class. Below is the data set. \begin{tabular}{|c|c|c|} \hline Gender & Number of Students & Birthplace \\ \hline \begin{tabular}{l} Male \\ Female \end{tabular} & \[ \begin{array}{c} 6 \\ 10 \end{array} \] & \begin{tabular}{l} San Diego \\ San Diego \end{tabular} \\ \hline \begin{tabular}{l} Male \\ Female \end{tabular} & \[ \begin{array}{c} 12 \\ 8 \end{array} \] & \begin{tabular}{l} Los \\ Angeles \\ Los \\ Angeles \end{tabular} \\ \hline \begin{tabular}{l} Male \\ Female \end{tabular} & \[ \begin{array}{l} 5 \\ 9 \end{array} \] & \begin{tabular}{l} San \\ Francisco \\ San \\ Francisco \end{tabular} \\ \hline \end{tabular} a. What is the probability that a student was born in Los Angeles given the student was male? (3 pts) b. What is the probability a student was born in San Diego or the student was female? (3 pts)
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Solution

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Solution Steps

Solution Approach

a. To find the probability that a student was born in Los Angeles given the student was male, we need to calculate the ratio of male students born in Los Angeles to the total number of male students.

b. To find the probability that a student was born in San Diego or is female, we use the formula for the probability of the union of two events: P(A or B) = P(A) + P(B) - P(A and B). Here, A is the event that a student was born in San Diego, and B is the event that a student is female.

Step 1: Calculate Total Students

We first determine the total number of male and female students: \[ \text{Total Male} = 6 + 12 + 5 = 23 \] \[ \text{Total Female} = 10 + 8 + 9 = 27 \] \[ \text{Total Students} = 23 + 27 = 50 \]

Step 2: Calculate Probability of Being Born in Los Angeles Given Male

To find the probability that a student was born in Los Angeles given the student was male, we use the formula: \[ P(\text{Los Angeles} | \text{Male}) = \frac{\text{Number of Male Students Born in Los Angeles}}{\text{Total Male Students}} = \frac{12}{23} \approx 0.5217 \]

Step 3: Calculate Probability of Being Born in San Diego or Being Female

We calculate the probability of being born in San Diego and the probability of being female: \[ P(\text{San Diego}) = \frac{6 + 10}{50} = \frac{16}{50} = 0.32 \] \[ P(\text{Female}) = \frac{27}{50} = 0.54 \] Next, we find the probability of being both born in San Diego and being female: \[ P(\text{San Diego} \cap \text{Female}) = \frac{10}{50} = 0.2 \] Now, we can use the formula for the union of two events: \[ P(\text{San Diego} \cup \text{Female}) = P(\text{San Diego}) + P(\text{Female}) - P(\text{San Diego} \cap \text{Female}) = 0.32 + 0.54 - 0.2 = 0.6600 \]

Final Answer

The answers to the questions are:

  • a. \( P(\text{Los Angeles} | \text{Male}) \approx 0.5217 \)
  • b. \( P(\text{San Diego} \cup \text{Female}) \approx 0.6600 \)

Thus, the final boxed answers are: \[ \boxed{P(\text{Los Angeles} | \text{Male}) \approx 0.5217} \] \[ \boxed{P(\text{San Diego} \cup \text{Female}) \approx 0.6600} \]

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