https://file.aiquickdraw.com/mathbot/file/1727298705900m5ik35l0.jpg

Let me break down the information provided in the table and calculate the necessary probabilities step by step.

### Table Summary:
- **Total individuals = 415**
- **Males: Right-handed = 204, Left-handed = 24**
- **Females: Right-handed = 160, Left-handed = 27**

Now, let's compute the probabilities:

1. **\( P(M) \)** (Probability of selecting a male):
   \[
   P(M) = \frac{\text{Number of males}}{\text{Total individuals}} = \frac{204 + 24}{415} = \frac{228}{415}
   \]
   \[
   P(M) = 0.5494
   \]

2. **\( P(F) \)** (Probability of selecting a female):
   \[
   P(F) = \frac{\text{Number of females}}{\text{Total individuals}} = \frac{160 + 27}{415} = \frac{187}{415}
   \]
   \[
   P(F) = 0.4506
   \]

3. **\( P(R) \)** (Probability of selecting a right-handed person):
   \[
   P(R) = \frac{\text{Number of right-handed individuals}}{\text{Total individuals}} = \frac{204 + 160}{415} = \frac{364}{415}
   \]
   \[
   P(R) = 0.8771
   \]

4. **\( P(L) \)** (Probability of selecting a left-handed person):
   \[
   P(L) = \frac{\text{Number of left-handed individuals}}{\text{Total individuals}} = \frac{24 + 27}{415} = \frac{51}{415}
   \]
   \[
   P(L) = 0.1229
   \]

5. **\( P(M \, \text{AND} \, R) \)** (Probability of selecting a male who is right-handed):
   \[
   P(M \, \text{AND} \, R) = \frac{\text{Number of right-handed males}}{\text{Total individuals}} = \frac{204}{415}
   \]
   \[
   P(M \, \text{AND} \, R) = 0.4916
   \]

6. **\( P(F \, \text{AND} \, L) \)** (Probability of selecting a female who is left-handed):
   \[
   P(F \, \text{AND} \, L) = \frac{\text{Number of left-handed females}}{\text{Total individuals}} = \frac{27}{415}
   \]
   \[
   P(F \, \text{AND} \, L) = 0.0651
   \]

7. **\( P(M \, \text{OR} \, L) \)** (Probability of selecting a male or a left-handed person):
   Use the formula:
   \[
   P(M \, \text{OR} \, L) = P(M) + P(L) - P(M \, \text{AND} \, L)
   \]
   Since \( P(M \, \text{AND} \, L) = \frac{24}{415} = 0.0578 \):
   \[
   P(M \, \text{OR} \, L) = 0.5494 + 0.1229 - 0.0578 = 0.6145
   \]

8. **\( P(F \, \text{OR} \, R) \)** (Probability of selecting a female or a right-handed person):
   Use the formula:
   \[
   P(F \, \text{OR} \, R) = P(F) + P(R) - P(F \, \text{AND} \, R)
   \]
   Since \( P(F \, \text{AND} \, R) = \frac{160}{415} = 0.3855 \):
   \[
   P(F \, \text{OR} \, R) = 0.4506 + 0.8771 - 0.3855 = 0.9422
   \]

9. **\( P(L \, | \, F) \)** (Conditional probability of being left-handed given the subject is female):
   Use the formula:
   \[
   P(L \, | \, F) = \frac{P(F \, \text{AND} \, L)}{P(F)}
   \]
   \[
   P(L \, | \, F) = \frac{0.0651}{0.4506} = 0.1445
   \]

Here is a summary of all the calculated probabilities:

- \( P(M) = 0.5494 \)
- \( P(F) = 0.4506 \)
- \( P(R) = 0.8771 \)
- \( P(L) = 0.1229 \)
- \( P(M \, \text{AND} \, R) = 0.4916 \)
- \( P(F \, \text{AND} \, L) = 0.0651 \)
- \( P(M \, \text{OR} \, L) = 0.6145 \)
- \( P(F \, \text{OR} \, R) = 0.9422 \)
- \( P(L \, | \, F) = 0.1445 \)

---

Would you like further clarification on any of the concepts? 

Here are 5 related questions:
1. How is the conditional probability formula derived?
2. What is the difference between "AND" and "OR" probabilities?
3. How can we interpret joint probabilities in real-world scenarios?
4. Can these probabilities help infer anything about the gender distribution?
5. What is the importance of total probability when calculating intersections?

**Tip:** Always remember, when dealing with "OR" probabilities, don't forget to subtract the overlap (i.e., when both events happen simultaneously).

The table below describes the distribution of a random sample of 415 individuals, organized by gender and whether they are right- or left-handed. Calculate various probabilities based on gender (M/F) and handedness (R/L).

Learn how to calculate various probabilities from a dataset of 415 individuals, organized by gender and handedness. This problem includes probability concepts such as joint probability, conditional probability, and the application of the Law of Total Probability. Suitable for students in Grades 10-12.

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