UpStudy Free Solution:

Step 1: Calculate the Total Number of Students

Total number of students = Number of males who passed + Number of males who failed + Number of females who passed + Number of females who failed

\(108 + 36 + 78 + 26 = 248\)

Step 2: Calculate the Probabilities

1. Probability of being male, \(P( \text { male} ) \):

\(P( \text { male} ) = \frac { \text { Number of males} } { \text { Total number of students} } = \frac { 108 + 36} { 248} = \frac { 144} { 248} \approx 0.58 \)

2. Probability of passing, \(P( \text { pass} ) \):

\(P( \text { pass} ) = \frac { \text { Number of students who passed} } { \text { Total number of students} } = \frac { 108 + 78} { 248} = \frac { 186} { 248} \approx 0.75\)

3. Probability of being male and passing, \(P( \text { male and pass} ) \):

\(P( \text { male and pass} ) = \frac { \text { Number of males who passed} } { \text { Total number of students} } = \frac { 108} { 248} \approx 0.435\)

Step 3: Calculate the Product of Individual Probabilities

\(P( \text { male} ) \times P( \text { pass} ) = 0.58 \times 0.75 \approx 0.435\)

Step 4: Compare the Probabilities

We compare \(P( \text { male} ) \times P( \text { pass} ) \) with \(P( \text { male and pass} ) \):

- \(P( \text { male} ) \times P( \text { pass} ) = 0.435\)

- \(P( \text { male and pass} ) = 0.435\)

Since these two probabilities are equal, the events are independent.

Key Concepts:

1. Probability: The measure of the likelihood of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

2. Independent Events: Two events are independent if the occurrence of one event does not affect the probability of the other. This can be checked by comparing the probability of the intersection of the events to the product of their individual probabilities.

Explanation:

- Calculating Total Number of Students: To find the probability of an event, first determine the total number of outcomes. For example, if you have the number of males and females who passed and failed a test, you sum these values to get the total number of students.

\(\text { Total number of students} = \text { Number of males who passed} + \text { Number of males who failed} + \text { Number of females who passed} + \text { Number of females who failed} \)

- Calculating Probabilities:

- Probability of Being Male:

\(P( \text { male} ) = \frac { \text { Number of males} } { \text { Total number of students} } \)

- Probability of Passing:

\(P( \text { pass} ) = \frac { \text { Number of students who passed} } { \text { Total number of students} } \)

- Probability of Being Male and Passing:

\(P( \text { male and pass} ) = \frac { \text { Number of males who passed} } { \text { Total number of students} } \)

- Product of Individual Probabilities:

- Calculate \(P( \text { male} ) \times P( \text { pass} ) \) to compare with \(P( \text { male and pass} ) \).

- Comparing Probabilities: If \(P( \text { male} ) \times P( \text { pass} ) \) equals \(P( \text { male and pass} ) \), then the events are independent.

Knowing each of these principles lets you use probability to establish whether events are or are not independent. Now, for all the types of problems where you need personalized support and detailed step-by-step solutions for your probability questions, use UpStudy Probability Solver. From the explanation to the final answer, UpStudy will make sure that you understand every topic of your class material. So go on and experience UpStudy today and make the most of enhanced learning!