Q.Two dice are thrown together. Let $A =$ 'the sum of the two numbers is $7$' and $B =$ 'the sum of the two numbers is $11$'. Are $A$ and $B$ mutually exclusive? Is the pair $\{A, B\}$ exhaustive for the sample space? Justify.

[!TLDR] Sum-7 and sum-11 outcomes cannot overlap, but together they cover far fewer than all 36 outcomes. [!ANSWER] $A …

For two dice, $A$ = 'sum is $7$' $=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$, so $n(A)=6$. $B$ = 'sum is $11$' $=\{(5,6),(6,5)\}$, so $n(B)=2$. Since a single throw has only one total, no outcome can simultaneously sum to both $7$ and $11$, so $A \cap B = \phi$ — mutually exclusive. But $n(A\cup B) = 6+2=8$, while $n(S)=36$; since $8 \ne 36$, $A \cup B \ne S$ (e.g. the outcome $(1,1)$, sum $2$, belongs to ne …

List or count the outcomes in each event, confirm no shared sum is possible (mutually exclusive), then compare $n(A …

A common mistake is assuming any two mutually exclusive events must automatically be exhaustive as well — mutual exclusivity only says they cannot …