Question

A coin is tossed three times. Consider the following events:
A: No head appears
B: Exactly one head appears
C: At least two heads appear
Which one of the following is correct?

• A. $(A\cup B)\cap (A\cup C)=B\cup C$
• B. $(A\cap B^{\prime })\cup (A\cap C^{\prime })=B^{\prime }\cup C^{\prime }$
• C. $A\cap B(B^{\prime }\cup C^{\prime })=A\cup B\cup C$
• D. $A\cap (B^{\prime }\cup C^{\prime })=B^{\prime }\cap C^{\prime }$

Answer 1

Answer: (D)

$A\cap (B^{\prime }\cup C^{\prime })=B^{\prime }\cap C^{\prime }$

$A=\left\{\right(T,T,T\left)\right\}$, $B=\left\{\right(H,T,T),(T,H,T),(T,T,H\left)\right\}$ and $C=\left\{\right(H,H,T),(T,H,H),(H,T,H),(H,H,H\left)\right\}$

$\therefore (A\cup B)\cap (A\cup C)$

$=\left\{\right(T,T,T),(H,T,T),(T,H,T),(T,T,H\left)\right\}\cap \left\{\right(T,T,T),(H,H,T),(T,H,H),(H,T,H),(H,H,H\left)\right\}$

$=\left\{\right(T,T,T\left)\right\}=A$

$\therefore (A\cap B^{\prime })\cup (A\cap C^{\prime })$

$=\left\{\right(H,H,T),(T,H,H),(H,T,H),(H,H,H\left)\right\}\cup \left\{\right(T,T,T),(H,T,T),(T,H,T),(T,T,H\left)\right\}=B^{\prime }\cup C^{\prime }$

$\therefore A\cap B\cap (B^{\prime }\cup C^{\prime })$

$=\left\{\right(T,T,T\left)\right\}\cap \left\{\right(H,T,T),(T,H,T),(T,T,H\left)\right\}\cap \left\{\right(T,T,T),(H,T,T)$

$,(T,H,T),(T,T,H),(H,H,T),(T,H,H),(H,T,H),(H,H,H)\}$

$=\varphi$

$\therefore A\cap (B^{\prime }\cup C^{\prime })$

$=\left\{\right(T,T,T\left)\right\}\cap \left\{\right(H,H,T),(T,H,H),(H,T,H),(H,H,H),(T,T,T),(H,T,T),(T,H,T),(T,T,H\left)\right\}$

$=\left\{\right(T,T,T\left)\right\}=A$