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 appears
Which of the following is/are true?

• A. $A,B$ and $C$ are exhaustive events
• B. $A$ and $C$ are mutually exclusive events
• C. $B$ and $C$ are mutually exclusive events
• D. $A,B$ and $C$ are pairwise mutually exclusive events

Answer 1

Answer: (A), (B), (C), (D)

$A,B$ and $C$ are pairwise mutually exclusive events

The sample space of the experiment is
$S=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}$
Events $A,B$ and $C$ are given by
$A=\left\{TTT\right\}B=\{HTT,THT,TTH\}C=\{HHT,HTH,THH,HHH\}$
Now,
$A\cup B\cup C=\{HHH,HHT,HTH,THH,HTT,THT,TTH,TTT\}=S$
Therefore, $A,B$ and $C$ are exhaustive events
Also, $A\cap B=\varphi ,B\cap C=\varphi$ and $A\cap C=\varphi$
Therefore, they are mutually exclusive.
Hence, $A,B$ and $C$ are pairwise mutually exclusive events