Q.Two fair dice are thrown. State the sample space and write the favorable outcomes for the following events. a) $A$: Sum of numbers on two dice is divisible by 3 or 4. b) $B$: Sum of numbers on two dice is 7. c) $C$: Odd number on the first die. d) $D$: Even number on the first die. e) Check whether events $A$ and $B$ are mutually exclusive and exhaustive. f) Check whether events $C$ and $D$ are mutually exclusive and exhaustive.

$S=\{(x,y): x,y\in\{1,\ldots,6\}\}$, $n(S)=36$.

a) Sum divisible by 3 (sums 3,6,9,12) or by 4 (sums 4,8,12): the div-by-3 outcomes number 12, the div-by-4 outcomes number 9, and they overlap only at sum 12 (the single outcome $(6,6)$), so by the addition-theorem count, $n(A)=12+9-1=20$.

b) Sum $=7$: outcomes $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$, so $n(B)=6$.

c) Odd number on first die (1,3,5), second die any value: $n(C)=3\times6=18$.

d) Even number on first die (2,4,6), second die any value: $n(D)=3\times6=18$.

e) A sum of 7 is never divisible by 3 or 4, so $A\cap B=\phi$: A and B are mutually exclusive. But $n(A)+n(B)=20+6=26\ne36$, so $A\cup B\ne S$: NOT exhaustive. …

Count each event's favourable outcomes directly (or via the addition-theorem overlap count for A), then test disjointness ($A …

Double-counting the overlap outcome $(6,6)$ when finding $n(A)$, or assuming mutually exclusive automatically impl …