A fair coin is tossed three times. Consider the events
A = {First toss is head}
B = {Second toss is head}
C = {Exactly two heads in a raw}
Which of the following statements is true?

Options

A. A and B are independent
B. A and B are not independent
C. A and C are independent
D. B and C are not independent

Answer

To solve the problem, we need to analyze the events A, B, and C based on the outcomes of tossing a fair coin three times.

### Step 1: Define the Sample Space
When a fair coin is tossed three times, the sample space consists of all possible outcomes. Each toss can result in either heads (H) or tails (T). Therefore, the total number of outcomes is:

\[ 2^3 = 8 \]

The sample space is:
\[ S = \{ HHH, HHT, HTH, HTT, THH, THT, TTH, TTT \} \]

### Step 2: Identify the Events
- Event A: The first toss is head.
\[ A = \{ HHH, HHT, HTH, HTT \} \quad \text{(4 outcomes)} \]

- Event B: The second toss is head.
\[ B = \{ HHH, HHT, THH, THT \} \quad \text{(4 outcomes)} \]

- Event C: Exactly two heads in a row.
\[ C = \{ HHT, HTH, THH \} \quad \text{(3 outcomes)} \]

### Step 3: Calculate Probabilities
- Probability of event A:
\[ P(A) = \frac{|A|}{|S|} = \frac{4}{8} = \frac{1}{2} \]

- Probability of event B:
\[ P(B) = \frac{|B|}{|S|} = \frac{4}{8} = \frac{1}{2} \]

- Probability of event C:
\[ P(C) = \frac{|C|}{|S|} = \frac{3}{8} \]

### Step 4: Check Independence of Events A and B
Two events A and B are independent if:
\[ P(A \cap B) = P(A) \times P(B) \]

- Find \( A \cap B \):
\[ A \cap B = \{ HHH, HHT \} \quad \text{(2 outcomes)} \]
\[ P(A \cap B) = \frac{2}{8} = \frac{1}{4} \]

- Check independence:
\[ P(A) \times P(B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]
Since \( P(A \cap B) = P(A) \times P(B) \), events A and B are independent.

### Step 5: Check Independence of Events A and C
- Find \( A \cap C \):
\[ A \cap C = \{ HHT, HTH \} \quad \text{(2 outcomes)} \]
\[ P(A \cap C) = \frac{2}{8} = \frac{1}{4} \]

- Check independence:
\[ P(A) \times P(C) = \frac{1}{2} \times \frac{3}{8} = \frac{3}{16} \]
Since \( P(A \cap C) \neq P(A) \times P(C) \), events A and C are not independent.

### Step 6: Check Independence of Events B and C
- Find \( B \cap C \):
\[ B \cap C = \{ HHT, THH \} \quad \text{(2 outcomes)} \]
\[ P(B \cap C) = \frac{2}{8} = \frac{1}{4} \]

- Check independence:
\[ P(B) \times P(C) = \frac{1}{2} \times \frac{3}{8} = \frac{3}{16} \]
Since \( P(B \cap C) \neq P(B) \times P(C) \), events B and C are not independent.

### Conclusion
From the analysis, we have:
1. Events A and B are independent.
2. Events A and C are not independent.
3. Events B and C are not independent.

### Final Statements
- Option 1: A and B are independent. **True**
- Option 2: A and B are not independent. **False**
- Option 3: A and C are independent. **False**
- Option 4: B and C are independent. **False**