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'.
Do they form a set of mutually exclusive and exhaustive events?

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Solution

Given: A coin is tossed three times.
$A :$ 'No head appears'
$B :$ 'Exactly one head appears'
$C :$ 'At least two heads appear'
The sample space is
$S = {\begin{matrix} H H H, H H T, H T H, T H H, H T T, T H T, T T H, T T T \end{matrix}}$
$A = {T T T}$
$B = {H T T, T H T, T T H}$
$C = {H H T, H T H, T H H, H H H}$

We need to find $A \cup B \cup C$ to check for exhaustive events
$A \cup B \cup C = {\begin{matrix} H H H, H H T, H T H, T H H, H T T, T H T, T T H, T T T \end{matrix}} = S$
Hence $A, B$ and $C$ are exhaustive events.

We need to find $A \cap B, B \cap C$ and $A \cap C$ for
checking mutually exclusive events.
$A \cap B = ϕ$ .
So, $A and B$ are mutually exclusive.
$B \cap C = ϕ$ .
So, $B and C$ are mutually exclusive.
$A \cap C = ϕ$ .
So, $A and C$ are mutually exclusive.
Hence, $A, B and C$ form a set of mutually exclusive and exhaustive events.

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Similar questions

A :

B :

C :

A is the event all four results are the same.

B is the event exactly one Head occurs.

C is the event at least two Heads occur.

P (A) + P (B) + P (C) > 1

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