https://file.aiquickdraw.com/mathbot/file/17291457544507zcxmoq7.jpg

Let's work through the problem step by step.

### Definitions:
- **Event A**: The selected student is a junior (for part (a)) or a senior (for part (b)).
- **Event B**: The selected student does not have a part-time job (for part (a)) or has a part-time job (for part (b)).

We need to determine if the events are independent by comparing the conditional probability \( P(B | A) \) to the marginal probability \( P(B) \).

The formula for conditional probability is:
\[
P(B|A) = \frac{P(A \cap B)}{P(A)}
\]
Where:
- \( P(A \cap B) \) is the joint probability of both events occurring.
- \( P(A) \) is the probability of event A.

### Part (a): Junior and No Part Time Job
1. **Event A**: The selected student is a junior.
   \[
   P(A) = \frac{132}{396} = 0.3333
   \]
2. **Event B**: The selected student does not have a part-time job.
   \[
   P(B) = \frac{195}{396} = 0.4924
   \]
3. **Joint Probability (A ∩ B)**: The student is both a junior and does not have a part-time job.
   \[
   P(A \cap B) = \frac{65}{396} = 0.1641
   \]
4. **Conditional Probability** \( P(B | A) \):
   \[
   P(B | A) = \frac{65}{132} = 0.4924
   \]
   
   Since \( P(B | A) = P(B) \), **the events are independent** for part (a).

### Part (b): Senior and Part Time Job
1. **Event A**: The selected student is a senior.
   \[
   P(A) = \frac{264}{396} = 0.6667
   \]
2. **Event B**: The selected student has a part-time job.
   \[
   P(B) = \frac{201}{396} = 0.5076
   \]
3. **Joint Probability (A ∩ B)**: The student is both a senior and has a part-time job.
   \[
   P(A \cap B) = \frac{134}{396} = 0.3384
   \]
4. **Conditional Probability** \( P(B | A) \):
   \[
   P(B | A) = \frac{134}{264} = 0.5076
   \]
   
   Since \( P(B | A) = P(B) \), **the events are independent** for part (b).

---

Do you need more details or have any questions?

Here are some relative questions for further practice:
1. What would happen if the probabilities \( P(B | A) \) and \( P(B) \) were not equal?
2. How do you calculate joint probability from a contingency table?
3. What is the difference between marginal and conditional probability?
4. How do you test for independence between two events?
5. Can two events be mutually exclusive and independent at the same time?

**Tip:** Always double-check your conditional and marginal probabilities to confirm whether events are independent or dependent!

The table shows the number of juniors and seniors at a high school who have part time jobs or do not have part time jobs. Determine whether events A and B are independent. Round your answers to four decimal places.

Understand how to determine if two events, such as being a junior or senior and having or not having a part-time job, are independent using conditional probability. Includes a step-by-step solution, calculations, and explanations. Suitable for Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation