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Let’s break down the problem part by part.

### Given table:
| Occupation    | Females | Males |
|---------------|---------|-------|
| Business      | 1200    | 1300  |
| IT            | 500     | 600   |
| Legal Service | 2000    | 2100  |
| Manufacturing | 750     | 800   |
| Hospitality   | 900     | 500   |
| Farming       | 650     | 700   |
| **Total**     | 6000    | 6000  |

### (a) **Joint Probability Table**  
A joint probability table shows the probability of different combinations of gender and occupation occurring. To calculate it, divide each cell in the table by the total number of workers (12,000 workers).

| Occupation    | Females | Males | Total |
|---------------|---------|-------|-------|
| Business      | \( \frac{1200}{12000} = 0.10 \)   | \( \frac{1300}{12000} = 0.1083 \) | \( 0.2083 \) |
| IT            | \( \frac{500}{12000} = 0.0417 \)  | \( \frac{600}{12000} = 0.05 \)    | \( 0.0917 \) |
| Legal Service | \( \frac{2000}{12000} = 0.1667 \) | \( \frac{2100}{12000} = 0.175 \)  | \( 0.3417 \) |
| Manufacturing | \( \frac{750}{12000} = 0.0625 \)  | \( \frac{800}{12000} = 0.0667 \)  | \( 0.1292 \) |
| Hospitality   | \( \frac{900}{12000} = 0.075 \)   | \( \frac{500}{12000} = 0.0417 \)  | \( 0.1167 \) |
| Farming       | \( \frac{650}{12000} = 0.0542 \)  | \( \frac{700}{12000} = 0.0583 \)  | \( 0.1125 \) |
| **Total**     | \( 0.50 \)     | \( 0.50 \)   | \( 1 \)    |

### (b) **Probability that a worker is male and works in hospitality**
Given the worker is male, what is the probability that he works in hospitality?

This is a conditional probability:  
\[ P(\text{Hospitality} | \text{Male}) = \frac{P(\text{Hospitality and Male})}{P(\text{Male})} \]

From the joint probability table:
- \( P(\text{Male}) = 0.50 \)
- \( P(\text{Hospitality and Male}) = 0.0417 \)

So,  
\[ P(\text{Hospitality} | \text{Male}) = \frac{0.0417}{0.50} = 0.0834 \]

### (c) **Probability that a worker is female and works in legal services**
Given the worker is female, what is the probability that she works in legal services?

This is also a conditional probability:
\[ P(\text{Legal Service} | \text{Female}) = \frac{P(\text{Legal Service and Female})}{P(\text{Female})} \]

From the joint probability table:
- \( P(\text{Female}) = 0.50 \)
- \( P(\text{Legal Service and Female}) = 0.1667 \)

So,  
\[ P(\text{Legal Service} | \text{Female}) = \frac{0.1667}{0.50} = 0.3334 \]

### (d) **Is occupation independent of gender?**
To test for independence, we check if \( P(\text{Occupation} | \text{Gender}) = P(\text{Occupation}) \) for all occupations. If true for all, then occupation and gender are independent.

As an example, for **hospitality**:
- \( P(\text{Hospitality}) = 0.1167 \)
- \( P(\text{Hospitality} | \text{Male}) = 0.0834 \)
- \( P(\text{Hospitality} | \text{Female}) = \frac{0.075}{0.50} = 0.15 \)

Since \( P(\text{Hospitality} | \text{Male}) \neq P(\text{Hospitality}) \) and \( P(\text{Hospitality} | \text{Female}) \neq P(\text{Hospitality}) \), **occupation is not independent of gender**.

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Would you like more details on any part? Here are 5 related questions for deeper exploration:

1. How do we calculate marginal probabilities from a joint probability table?
2. What is the difference between conditional probability and joint probability?
3. How can we verify independence using conditional probabilities?
4. How are joint and marginal probabilities related?
5. Can you explain how gender distribution affects probability results in this table?

**Tip:** When solving probability problems, organizing your data into tables like joint and marginal probability tables can make interpreting results much easier.

The California Department of Labor surveyed the relationship between gender and occupation. The data shows the number of female and male workers in different occupational categories, and asks to solve for joint probabilities, conditional probabilities, and test for independence.

Solve joint and conditional probability problems involving gender and occupation data. The question involves calculating joint probabilities, conditional probabilities, and testing for independence using real-world labor statistics.

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