Consider a random experiment where two fair coins are tossed. Let $A$ be the event that denotes HEAD on both the throws, $B$ be the event that denotes HEAD on the first throw, and $C$ be the event that denotes HEAD on the second throw. Which of the following statements is/are TRUE?

Let the sample space for tossing two fair coins be:

with each elementary outcome having a probability of $\frac{1}{4}$.

The events are defined as follows:

• Event $A$ (HEAD on both throws): $A = \{HH\} \implies P(A) = \frac{1}{4}$

• Event $B$ (HEAD on the first throw): $B = \{HH, HT\} \implies P(B) = \frac{2}{4} = \frac{1}{2}$

• Event $C$ (HEAD on the second throw): $C = \{HH, TH\} \implies P(C) = \frac{2}{4} = \frac{1}{2}$

Now we test each statement:

1. Option A: $A$ and $B$ are independent.

   $$A \cap B = \{HH\} \implies P(A \cap B) = \frac{1}{4}$$

   $$P(A) \cdot P(B) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \neq P(A \cap B)$$

   Hence, $A$ and $B$ are not independent. Option A is FALSE.

2. Option B: $A$ and $C$ are independent.

   $$A \cap C = \{HH\} \implies P(A \cap C) = \frac{1}{4}$$

   $$P(A) \cdot P(C) = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \neq P(A \cap C)$$

   Hence, $A$ and $C$ are not independent. Option B is FALSE.

3. Option C: $B$ and $C$ are independent.

   $$B \cap C = \{HH\} \implies P(B \cap C) = \frac{1}{4}$$

   $$P(B) \cdot P(C) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = P(B \cap C)$$

   Hence, $B$ and $C$ are independent. Option C is TRUE.

4. Option D: $\text{Prob}(B|C) = \text{Prob}(B)$.

   $$\text{Prob}(B|C) = \frac{P(B \cap C)}{P(C)} = \frac{1/4}{1/2} = \frac{1}{2} = \text{Prob}(B)$$

   This is also an immediate property of independent events $B$ and $C$. Option D is TRUE.

Thus, statements C and D are correct.