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An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
$A:$ the sum is greater than $8$,
$B : 2$ occurs on either die
$C:$ the sum is at least $ 7$ and a multiple of $3.$
Which pairs of these events are mutually exclusive ?
Solution
When a pair of dice is rolled, the sample space is given by
$S=\{(x, y): x$, $y=1,2,3,4,5,6\}$
$=\left\{\begin{array}{l}(1,1),(1,2),(1,3),(1,4),(1,5),(1,6) \\ (2,1),(2,2),(2,3),(2,4),(2,5),(2,6) \\ (3,1),(3,2),(3,3),(3,4),(3,5),(3,6) \\ (4,1),(4,2),(4,3),(4,4),(4,5),(4,6) \\ (5,1),(5,2),(5,3),(5,4),(5,5),(5,6) \\ (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\end{array}\right\}$
Accordingly,
$A=\{(3,6),(4,5),(4,6)$, $(5,4),(5,5),$ $(5,6)(6,3)$, $(6,4),(6,5),(6,6)\}$
$B =\{(2,1),(2,2),(2,3),$ $(2,4),(2,5),$ $(2,6)(1,2),(3,2)$, $(4,2),(5,2),(6,2)\}$
$C=\{(3,6),(4,5),(5,4),(6,3),(6,6)\}$
It is observed that $A \cap B=\phi$
$B \cap C=\phi$
$C \cap A=\{(3,6),(4,5)$, $(5,4),(6,3),(6,6)\}$ $ \neq \phi$
Hence, events $A$ and $B$ and events $B$ and $C$ are mutually exclusive.
