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 $\le 5$
State true or false : (give reason for your answer)
(i) $A$ and $B$ are mutually exclusive
(ii) $A$ and $B$ are mutually exclusive and exhaustive
(iii) $A=B^{\prime }$
(iv) $A$ and $C$ are mutually exclusive
(v) $A$ and $B^{\prime }$ are mutually exclusive
(vi) $A^{\prime },B^{\prime },C$ are mutually exclusive and exhaustive

Answer 1

$S=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(2,1),(2,2),...........,(2,6\left)\right(3,1),(3,2)..............(3,6\left)\right(4,1),(4,2).............(4,6),(5,1),(5,2).............(5,6\left)\right(6,1),(6,2)...........(6,6\left)\right\}$
$A=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
$B=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right\}$
$C=\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1\left)\right\}$
(i) Clearly, $A\cap B=\varphi$
$\therefore$ $A$ and $B$ are mutually exclusive.
Thus the given statement is true.
(ii)$A\cap B=\varphi$ and $A\cup B=S$
$\therefore$ $A$ and $B$ are mutually exclusive and exhaustive.
Thus the given statement is true.
(iii)$B^{\prime }=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}=A$
Thus the given statement is true.
(iv) $A\cap C=\left\{\right(2,1),(2,2),(2,3),(4,1\left)\right\}\ne \varphi$
$\therefore$ $A$ and $C$ are not mutually exclusive.
Thus the given statement is false..
(v) $A\cap B^{\prime }=A\cap A=A$
$\therefore A\cap B^{\prime }\ne \varphi$
$\therefore$ A and B' are not mutually exclusive.
Thus the given statement is false.
(vi) $A^{\prime }\cup B^{\prime }\cup C=S$
However $B^{\prime }\cap C=\left\{\right(2,1),(2,2),(2,3),(4,1\left)\right\}\ne \varphi$
Therefore events $A^{\prime },B^{\prime }$ and $C$ are not mutually exclusive and exhaustive.
Thus the given statement is false.