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
Describe the events
(i) A'
(ii) not B
(iii) A or B
(iv) A and B
(v) A but not C
(vi) B or C
(vii) B and C
(viii) $A\cap B^{\prime }\cap C^{\prime }$

Answer 1

When two dice are thrown then
$S=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6\left)\right(3,1),(3,2),(3,3),(3,4),(3,5),(3,6\left)\right(4,1),(4,2),(4,3),(4,4),(4,5),(4,6\left)\right(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
A: Getting an even number on the first die.
$=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6\left)\right(4,1),(4,2),(4,3),(4,4),(4,5),(4,6\left)\right(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
B: getting an odd number on the first die
$\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(3,1),(3,2),(3,3),(3,4),(3,5),(3,6\left)\right(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right\}$
C: getting the sum of the numbers on the dice $\le$ 5
$\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,2),(4,1\left)\right\}$
(i) A' = $\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(3,1),(3,2),(3,3),(3,4),(3,5),(3,6\left)\right(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right\}$
(ii) Not $B=\left\{\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6\left)\right(4,1),(4,2),(4,3),(4,4),(4,5),(4,6\left)\right(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
(iii) A or B =
$A\cup B=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6\left)\right(2,1),(2,2),(2,3),(2,4),(2,5),(2,6\left)\right(3,1),(3,2),(3,3),(3,4),(3,5),(3,6\left)\right(4,1),(4,2),(4,3),(4,4),(4,5),(4,6\left)\right(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$
(iv) $AandB=A\cap B=\varphi$
(v) $Abutnot(C=A-C=\{(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)\}$
(vi)$BorC=B\cap C=\left\{\right(1,1),(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4),(3,5),(3,6),(4,1),(5,1),(5,2),(5,3),(5,4),(5,5),(5,6\left)\right\}$
(viii) $BandC=B\cap C=\{1,1),(1,2),(1,3),(1,4),(3,1),(3,2)\}$
(viii) $A\cap B^{\prime }\cap C^{\prime }=\left\{\right(2,4),(2,5),(2,6),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6\left)\right\}$