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Mutually Exclusive Events

Two dice are ...

Question

Two dice are thrown. The events $A, B and C$ are as follows:
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
$C :$ getting the sum of the numbers on the dice $\leq 5$

$i .$ State true or false: (give reason for your answer)
$A$ and $B$ are mutually exclusive.
$i i .$ State true or false: (give reason for your answer)
$A$ and $B$ are mutually exclusive and exhaustive.
$i i i .$ State true or false: (give reason for your answer)
$A = B'$ .
$i v .$ State true or false: (give reason for your answer)
$A$ and $C$ are mutually exclusive.
$v .$ State true or false: (give reason for your answer)
$A$ and $B^{'}$ are mutually exclusive.
$v i .$ State true or false: (give reason for your answer)
$A', B', C$ are mutually exclusive and exhaustive.

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Solution

$(i)$ Given : Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
"If two dice are thrown, then the sample space is"
$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$
$\Rightarrow B = S - A = A^{'}$
$∴ A \cap B = A \cap A^{'} = ϕ$
Hence, $A, B$ are mutually exclusive events.
$∴$ Statement is true.

$(i i)$ Given: Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
If two dice are thrown, then the sample space is

$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$
$\Rightarrow B = S - A = A^{'}$
$∴ A \cap B = A \cap A^{'} = ϕ$
So, $A, B$ are mutually exclusive events.
Also,
$A \cup B = A \cup A' = S$
So, $A$ $B$ are exhaustive
Hence, $A$ and $B$ are mutually exclusive and exhaustive
$∴$ Statement is ture.

$(i i i)$ Given : Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
If two dice are thrown, then the sample space is

$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$
$B^{'} = S - B$
$= ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ = A$
Hence, $B^{'} = A$
$∴$ Statement is true.

$(i v)$ Given: Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
$C :$ getting the sum of the numbers on the dice $\leq 5$
If two dice are thrown, then the sample space is

$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$C :$ getting the sum of the numbers on the dice $\leq 5$
$C = {\begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1) \end{matrix}}$

$A \cap C = {(2, 1), (2, 2), (2, 3), (4, 1)$
As $A \cap C \neq ϕ$ , so $A$ and $C$ are not mutually exclusive.
$∴$ Satement is false.

$(v)$ Given: Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
If two dice are thrown, then the sample space is

$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B^{'} = S - B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ = A$
Therefore,
$= A \cap B' = A \cap A = A \neq ϕ$
Hence, $A$ and $B$ ′ are not mutually exclusive.
$∴$ Statement is false.

$(v i)$ Given : Two dice are thrown
$A :$ getting an even number on the first die.
$B :$ getting an odd number on the first die.
$C :$ getting the sum of the numbers on the dice $\leq 5$
If two dice are thrown, then the sample space is

$S = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭$

$A = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$B = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭$

$C = {\begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1) \end{matrix}}$

$A^{'} = S - A$
$= ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ = B$

$B^{'} = S - B$
$= ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭ = A$

$\Rightarrow B^{'} = A$
$\Rightarrow A^{'} = B$

$C = {\begin{matrix} (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1) \end{matrix}}$

$A' \cap B' = B \cap A = ϕ$
So, $A^{'}$ and $B^{'}$ are mutually exclusive.
$B^{'} \cap C = A \cap C \neq ϕ$
So, $B'$ and $C$ are not mutually exclusive.
$A' \cap C = B \cap C \neq ϕ$
So, $A'$ and $C$ are not mutually exclusive.
$A' \cup B' \cup C = A' \cup A \cup C = S \cup C = S$
Hence, $A', B', C$ are not mutually exclusive but exhaustive.
$∴$ Statement is false.

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