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Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us consider the following events associated with this experiment
$A:$ $^{\prime}$ the sum is even $^{\prime}$.
$B:$ $^{\prime}$the sum is a multiple of $3$$^{\prime}$
$C:$ $^{\prime}$the sum is less than $4 $$^{\prime}$
$D:$ $^{\prime}$the sum is greater than $11$$^{\prime}$.
Which pairs of these events are mutually exclusive ?
$A$ and $B$
$A$ and $D$
$B$ and $D$
$C$ and $D$
Solution
Solution There are 36 elements in the sample space $S=\{(x, y): x, y=1,2,3,4,5,6\}$ Then
$A =\{(1,1),(1,3),(1,5),(2,2),$ $(2,4),(2,6),$ $(3,1),(3,3),$ $(3,5),(4,2),(4,4),$ $(4,6),(5,1),(5,3),(5,5),(6,2),$ $(6,4),(6,6)\} $
$B =\{(1,2),(2,1),(1,5),$ $(5,1),(3,3),$ $(2,4),(4,2),(3,6),$ $(6,3),(4,5),(5,4),(6,6)\}$
$C=\{(1,1),(2,1),(1,2)\}$ and $D=\{(6,6)\}$
We find that
$A \cap B=\{(1,5),(2,4),(3,3),(4,2),(5,1),(6,6)\} \neq \phi$
Therefore, $A$ and $B$ are not mutually exclusive events.
Similarly $A \cap C \neq \phi, $ $A \cap D \neq \phi, $ $B \cap C \neq \phi$ and $B \cap D \neq \phi$
Thus, the pairs of events, $(A, C),(A, D),(B, C),(B, D)$ are not mutually exclusive events.
Also $C \cap D =\phi$ and so $C$ and $D$ are mutually exclusive events.
