Question 53

Sample space for two fair coin tosses: $S = \{HH, HT, TH, TT\}$, each with probability $1/4$.

• Event $A$ (HEAD on both throws): $A = \{HH\} \implies P(A) = 1/4$
• Event $B$ (HEAD on first throw): $B = \{HH, HT\} \implies P(B) = 2/4 = 1/2$
• Event $C$ (HEAD on second throw): $C = \{HH, TH\} \implies P(C) = 2/4 = 1/2$

Testing independence:

1. For $B$ and $C$:

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

   Hence, $B$ and $C$ are independent (Statement C is TRUE).

2. Conditional probability $\text{Prob}(B \mid C)$:

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

   Hence, Statement D is TRUE.

3. For $A$ and $B$: $A \cap B = \{HH\} \implies P(A \cap B) = 1/4$, while $P(A) P(B) = (1/4)(1/2) = 1/8 \neq 1/4$. Thus $A$ and $B$ are dependent (Statement A is FALSE, and by symmetry Statement B is FALSE).