Question

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

• A. $A$ and $B$ are independent.
• B. $A$ and $C$ are independent.
• C. $B$ and $C$ are independent.
• D. $\Pr(B \mid C) = \Pr(B)$.

Hint:

To test independence: compute $\Pr(X\cap Y)$ and compare with $\Pr(X)\Pr(Y)$. Coin toss events for different coins are independent, but compound events (like $A$) may not be independent of single tosses.

Answer 1

The Correct Option is C

Step 1: Sample space.
Two fair coins tossed: $S=\{HH, HT, TH, TT\}$. Each outcome has probability $1/4$.
Step 2: Define events.
- $A=\{HH\}$, so $\Pr(A)=1/4$.
- $B=\{HH, HT\}$, so $\Pr(B)=1/2$.
- $C=\{HH, TH\}$, so $\Pr(C)=1/2$.
Step 3: Check independence of $A$ and $B$.
$\Pr(A\cap B)=\Pr(\{HH\})=1/4$.
$\Pr(A)\Pr(B)=(1/4)(1/2)=1/8$.
Since $1/4\neq 1/8$, $A$ and $B$ are not independent. (A) False.
Step 4: Check independence of $A$ and $C$.
$\Pr(A\cap C)=\Pr(\{HH\})=1/4$.
$\Pr(A)\Pr(C)=(1/4)(1/2)=1/8$.
Since $1/4\neq 1/8$, $A$ and $C$ are not independent. (B) False.
Step 5: Check independence of $B$ and $C$.
$\Pr(B\cap C)=\Pr(\{HH\})=1/4$.
$\Pr(B)\Pr(C)=(1/2)(1/2)=1/4$.
Since equal, $B$ and $C$ are independent. (C) True.
Step 6: Check conditional probability $\Pr(B\mid C)$.
$\Pr(B\mid C)=\frac{\Pr(B\cap C)}{\Pr(C)}=\frac{1/4}{1/2}=1/2$.
But $\Pr(B)=1/2$. Thus $\Pr(B\mid C)=\Pr(B)$. (D) True.
\[ \boxed{\text{True statements: (C) and (D)}} \]