Hyperbola Question - None

Question

Two dice are thrown independently. Let A be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, B be the event that the number appeared on the $1^{\text {st }}$ die is even and that on the second die is odd, and C be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then:

Accepted Answer

Here,
${A}=\{(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),$ (2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)}
$ herefore n({~A})=15$
${B}=\{(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),$ (6,1),(6,3),(6,5)}
$ herefore n({~B})=9$
${C}=\{(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),$ (5,2),(5,4),(5,6)}
$ herefore n(c)=9$

False, $n({~A})=15, n({~B})=9 eq 6, n({c})=9 eq 6$
$({A} \cap {C})=\{(1,2),(1,4),(1,6),(3,4),(3,6),(5,6)\}$
$ herefore n({~A} \cap {C})=6$
$({B} \cap {C})=\phi$
$ herefore n({~B} \cap {C})=0$
and $n(({~A} \cup {~B}) \cap {C})=6$
${P}({B})=\frac{9}{36}=\frac{1}{4}, {P}({C})=\frac{9}{36}=\frac{1}{4}, {P}({B} \cap {C})=0$
As, ${P}({B}) \cdot {P}({C}) eq {P}({B} \cap {C})$
So, B and C are not independent.
As, ${A} \cap {B}=\{(4,5)\} eq \phi$
So, A and B are not exclusive events.

Question:

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