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Let's analyze the problem based on the data provided. Here’s a step-by-step breakdown:

### Given Data:
The table shows a sample distribution of 415 individuals by gender and handedness:

| Handedness by Gender | Right-handed | Left-handed | Total |
|----------------------|--------------|-------------|-------|
| Male                 | 204          | 24          | 228   |
| Female               | 160          | 27          | 187   |
| **Total**            | 364          | 51          | 415   |

The random variables are defined as:
- \( M \): Subject is male
- \( F \): Subject is female
- \( R \): Subject is right-handed
- \( L \): Subject is left-handed

### We need to calculate the following probabilities:

#### 1. \( P(M \text{ OR } L) \)
We use the formula:
\[
P(M \text{ OR } L) = P(M) + P(L) - P(M \text{ AND } L)
\]
From the table:
- \( P(M) = \frac{228}{415} \approx 0.5494 \)
- \( P(L) = \frac{51}{415} \approx 0.1229 \)
- \( P(M \text{ AND } L) = \frac{24}{415} \approx 0.0578 \)

Substituting these values:
\[
P(M \text{ OR } L) = 0.5494 + 0.1229 - 0.0578 = 0.6145
\]

#### 2. \( P(F \text{ OR } R) \)
We use the formula:
\[
P(F \text{ OR } R) = P(F) + P(R) - P(F \text{ AND } R)
\]
From the table:
- \( P(F) = \frac{187}{415} \approx 0.4506 \)
- \( P(R) = \frac{364}{415} \approx 0.8771 \)
- \( P(F \text{ AND } R) = \frac{160}{415} \approx 0.3855 \)

Substituting these values:
\[
P(F \text{ OR } R) = 0.4506 + 0.8771 - 0.3855 = 0.9422
\]

#### 3. \( P(M \text{ OR } F) \)
Since every subject is either male or female:
\[
P(M \text{ OR } F) = P(M) + P(F) = 0.5494 + 0.4506 = 1
\]

### Summary of results:
- \( P(M \text{ OR } L) = 0.6145 \)
- \( P(F \text{ OR } R) = 0.9422 \)
- \( P(M \text{ OR } F) = 1 \)

Would you like more details or have any questions?

### 5 Related Questions
1. What is the probability of selecting a left-handed male?
2. How do you calculate conditional probabilities like \( P(F | R) \)?
3. How would the probabilities change if there were more left-handed females?
4. Can you explain why \( P(M \text{ OR } F) = 1 \)?
5. What would \( P(R | F) \) represent, and how do you calculate it?

### Tip
Always ensure that the sum of probabilities across mutually exclusive and collectively exhaustive events equals 1.

One subject from the sample is selected at random. Let M = the subject is male, F = the subject is female, R = the subject is right-handed, and L = the subject is left-handed. Given a table showing the distribution of handedness by gender for 415 individuals, calculate various probabilities such as P(M OR L), P(F OR R), and P(M OR F).

Learn how to calculate probabilities for a random sample of 415 individuals based on their handedness and gender. This problem involves the use of the addition rule for probabilities and conditional probability, with detailed solutions for P(M OR L), P(F OR R), and P(M OR F).

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