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The image you uploaded shows a probability table and a set of probability-related questions involving social media usage and opinions about television programs, categorized by gender.

### Explanation of the Table:
- The table contains data for both **Female** and **Male** respondents and whether they **Use Social Media** or **Do Not Use Social Media** to voice opinions about television programs.
- The values provided are joint probabilities, representing the probability that a respondent falls into one of the four possible groups: Female using social media, Female not using social media, Male using social media, and Male not using social media.
  
### Questions:
1. **(a) Joint Probability Table:**
   - The table is already filled with the joint probabilities for each category.
     - \( P(\text{Female and Uses Social Media}) = 0.2933 \)
     - \( P(\text{Male and Uses Social Media}) = 0.2403 \)
     - \( P(\text{Female and Doesn't Use Social Media}) = 0.2030 \)
     - \( P(\text{Male and Doesn't Use Social Media}) = 0.2634 \)
     - Total probabilities for using and not using social media are also provided.

2. **(b) What is the probability a respondent is female?**
   - Answer: 0.4963. This is derived from the sum of probabilities of females using and not using social media.

3. **(c) What is the conditional probability a respondent uses social media given the respondent is female?**
   - The answer appears to be \( P(\text{Uses Social Media} | \text{Female}) = 0.5336 \).

4. **(d) Are events \( F \) (Female) and \( A \) (Using Social Media) independent?**
   - This asks if knowing whether a respondent is female affects the probability of them using social media.
   - Independence would mean that:
     \[
     P(A | F) = P(A)
     \]
     Where \( P(A) \) is the probability of using social media (without considering gender).
   - Based on the calculations and provided answer, the user concludes that the events are independent because \( P(A|F) = P(A) = 0.5336 \).

### Do you need help verifying the calculations or further clarification? Let me know!

Here are five related questions to expand on this topic:
1. How do we calculate the probability of being male using this table?
2. What is the conditional probability of not using social media given the respondent is male?
3. How can we calculate the joint probability of being female and not using social media?
4. What does it mean for two events to be dependent in probability theory?
5. How do we compute conditional probabilities when multiple conditions are involved?

**Tip:** When working with joint probability tables, always check the marginal totals to ensure that all probabilities sum to 1.

Based on a survey of 1,340 self-identifying females and males who were asked if they use social media to voice their opinions about television programs, analyze the probabilities and relationships between gender and social media usage.

Explore a probability analysis of social media usage by gender based on a survey of 1,340 individuals. Learn how to calculate joint probabilities, conditional probabilities, and assess the independence of events using detailed step-by-step solutions. This analysis is suitable for college-level students studying probability and statistics.

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