A family has three children. Event `'A'` is that family has at most one boy, Event `'B'` is that family has at least one boy and one girl, Event `'C'` is that the family has at most one girl. Then

Options

A. Events `'A'` and `'B'` are independent
B. Events `'A'` and `'B'` are notindependent
C. Events `A,B,C` are independent
D. Events `A,B,C` are not independent

Explanation

To solve the problem step by step, we will first define the events and then calculate the probabilities associated with each event.

### Step 1: Define the Sample Space
The sample space for a family with three children can be represented as:
- BBB (3 boys)
- BBG (2 boys, 1 girl)
- BGB (2 boys, 1 girl)
- GBB (2 boys, 1 girl)
- BGG (1 boy, 2 girls)
- GBG (1 boy, 2 girls)
- GGB (1 boy, 2 girls)
- GGG (3 girls)

Thus, the sample space \( S \) consists of 8 outcomes:
\[ S = \{ BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG \} \]

### Step 2: Define Events A, B, and C
- **Event A**: The family has at most one boy.
- Outcomes: GGG, BGG, GBG, GGB (4 outcomes)

- **Event B**: The family has at least one boy and one girl.
- Outcomes: BBG, BGB, GBB, BGG, GBG, GGB (6 outcomes)

- **Event C**: The family has at most one girl.
- Outcomes: BBB, BBG, BGB, GBB (4 outcomes)

### Step 3: Calculate Probabilities
- Total outcomes in the sample space \( |S| = 8 \)

1. **Probability of Event A**:
\[ P(A) = \frac{\text{Number of outcomes in A}}{|S|} = \frac{4}{8} = \frac{1}{2} \]

2. **Probability of Event B**:
\[ P(B) = \frac{\text{Number of outcomes in B}}{|S|} = \frac{6}{8} = \frac{3}{4} \]

3. **Probability of Event C**:
\[ P(C) = \frac{\text{Number of outcomes in C}}{|S|} = \frac{4}{8} = \frac{1}{2} \]

### Step 4: Calculate Intersection of Events
1. **Intersection of A and B**:
- Outcomes in \( A \cap B \): BBG, BGB, GBB (3 outcomes)
\[ P(A \cap B) = \frac{3}{8} \]

2. **Intersection of A and C**:
- Outcomes in \( A \cap C \): GGG (1 outcome)
\[ P(A \cap C) = \frac{1}{8} \]

3. **Intersection of B and C**:
- Outcomes in \( B \cap C \): BBG, BGB, GBB (3 outcomes)
\[ P(B \cap C) = \frac{3}{8} \]

### Step 5: Check Independence
1. **Check if A and B are independent**:
- \( P(A) \times P(B) = \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \)
- Since \( P(A \cap B) = \frac{3}{8} \), A and B are independent.

2. **Check if A and C are independent**:
- \( P(A) \times P(C) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \)
- Since \( P(A \cap C) = \frac{1}{8} \), A and C are not independent.

3. **Check if B and C are independent**:
- \( P(B) \times P(C) = \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \)
- Since \( P(B \cap C) = \frac{3}{8} \), B and C are independent.

### Conclusion
- Events A and B are independent.
- Events A and C are not independent.
- Events B and C are independent.

### Final Answer
Thus, the correct conclusion is that events A and B are independent, while A and C are not independent, and B and C are independent.